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"[W]e think of the analysis of variance as a way of understanding and structuring multilevel models—not as an alternative to regression but as a tool for summarizing complex high-dimensional inferences ..."
For a single factor
The simplest experiment suitable for ANOVA analysis is the completely randomized experiment with a single factor. More complex experiments with a single factor involve constraints on randomization and include completely randomized blocks and Latin squares (and variants: Graeco-Latin squares, etc.). The more complex experiments share many of the complexities of multiple factors. A relatively complete discussion of the analysis (models, data summaries, ANOVA table) of the completely randomized experiment is available. There are some alternatives to conventional one-way analysis of variance, e.g. | [
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: Welch's heteroscedastic F test, Welch's heteroscedastic F test with trimmed means and Winsorized variances, Brown-Forsythe test, Alexander-Govern test, James second order test and Kruskal-Wallis test, available in onewaytests R
It is useful to represent each data point in the following form, called a statistical model:
where
i = 1, 2, 3, …, R
j = 1, 2, 3, …, C
μ = overall average (mean)
τj = differential effect (response) associated with the j level of X; this assumes that overall the values of τj add to zero (that is, )
εij = noise or error associated with the particular ij data value
That is, we envision an additive model that says every data point can be represented by summing three quantities: the true mean, averaged over all factor levels being investigated, plus an incremental component associated with the particular column (factor level), plus a final component associated with everything else affecting that specific data value. For multiple factors
ANOVA generalizes to the study of the effects of multiple factors. When the experiment includes observations at all combinations of levels of each factor, it is termed factorial. Factorial experiments are more efficient than a series of single factor experiments and the efficiency grows as the number of factors increases. Consequently, factorial designs are heavily used. The use of ANOVA to study the effects of multiple factors has a complication. | [
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In a 3-way ANOVA with factors x, y and z, the ANOVA model includes terms for the main effects (x, y, z) and terms for interactions (xy, xz, yz, xyz). All terms require hypothesis tests. The proliferation of interaction terms increases the risk that some hypothesis test will produce a false positive by chance. Fortunately, experience says that high order interactions are rare. The ability to detect interactions is a major advantage of multiple factor ANOVA. Testing one factor at a time hides interactions, but produces apparently inconsistent experimental results. Caution is advised when encountering interactions; Test interaction terms first and expand the analysis beyond ANOVA if interactions are found. Texts vary in their recommendations regarding the continuation of the ANOVA procedure after encountering an interaction. Interactions complicate the interpretation of experimental data. Neither the calculations of significance nor the estimated treatment effects can be taken at face value. "A significant interaction will often mask the significance of main effects." Graphical methods are recommended to enhance understanding. Regression is often useful. A lengthy discussion of interactions is available in Cox (1958). Some interactions can be removed (by transformations) while others cannot. A variety of techniques are used with multiple factor ANOVA to reduce expense. One technique used in factorial designs is to minimize replication (possibly no replication with support of analytical trickery) and to combine groups when effects are found to be statistically (or practically) insignificant. An experiment with many insignificant factors may collapse into one with a few factors supported by many replications. | [
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Associated analysis
Some analysis is required in support of the design of the experiment while other analysis is performed after changes in the factors are formally found to produce statistically significant changes in the responses. Because experimentation is iterative, the results of one experiment alter plans for following experiments. Preparatory analysis
The number of experimental units
In the design of an experiment, the number of experimental units is planned to satisfy the goals of the experiment. Experimentation is often sequential. Early experiments are often designed to provide mean-unbiased estimates of treatment effects and of experimental error. Later experiments are often designed to test a hypothesis that a treatment effect has an important magnitude; in this case, the number of experimental units is chosen so that the experiment is within budget and has adequate power, among other goals. Reporting sample size analysis is generally required in psychology. "Provide information on sample size and the process that led to sample size decisions." The analysis, which is written in the experimental protocol before the experiment is conducted, is examined in grant applications and administrative review boards. Besides the power analysis, there are less formal methods for selecting the number of experimental units. These include graphical methods based on limiting the probability of false negative errors, graphical methods based on an expected variation increase (above the residuals) and methods based on achieving a desired confidence interval. Power analysis
Power analysis is often applied in the context of ANOVA in order to assess the probability of successfully rejecting the null hypothesis if we assume a certain ANOVA design, effect size in the population, sample size and significance level. | [
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Power analysis can assist in study design by determining what sample size would be required in order to have a reasonable chance of rejecting the null hypothesis when the alternative hypothesis is true. Effect size
Several standardized measures of effect have been proposed for ANOVA to summarize the strength of the association between a predictor(s) and the dependent variable or the overall standardized difference of the complete model. Standardized effect-size estimates facilitate comparison of findings across studies and disciplines. However, while standardized effect sizes are commonly used in much of the professional literature, a non-standardized measure of effect size that has immediately "meaningful" units may be preferable for reporting purposes. Model confirmation
Sometimes tests are conducted to determine whether the assumptions of ANOVA appear to be violated. Residuals are examined or analyzed to confirm homoscedasticity and gross normality. Residuals should have the appearance of (zero mean normal distribution) noise when plotted as a function of anything including time and
modeled data values. Trends hint at interactions among factors or among observations. Follow-up tests
A statistically significant effect in ANOVA is often followed by additional tests. This can be done in order to assess which groups are different from which other groups or to test various other focused hypotheses. Follow-up tests are often distinguished in terms of whether they are "planned" (a priori) or "post hoc." Planned tests are determined before looking at the data, and post hoc tests are conceived only after looking at the data (though the term "post hoc" is inconsistently used). | [
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The follow-up tests may be "simple" pairwise comparisons of individual group means or may be "compound" comparisons (e.g., comparing the mean pooling across groups A, B and C to the mean of group D). Comparisons can also look at tests of trend, such as linear and quadratic relationships, when the independent variable involves ordered levels. Often the follow-up tests incorporate a method of adjusting for the multiple comparisons problem. Study designs
There are several types of ANOVA. Many statisticians base ANOVA on the design of the experiment, especially on the protocol that specifies the random assignment of treatments to subjects; the protocol's description of the assignment mechanism should include a specification of the structure of the treatments and of any blocking. It is also common to apply ANOVA to observational data using an appropriate statistical model. Some popular designs use the following types of ANOVA:
One-way ANOVA is used to test for differences among two or more independent groups (means), e.g. different levels of urea application in a crop, or different levels of antibiotic action on several different bacterial species, or different levels of effect of some medicine on groups of patients. However, should these groups not be independent, and there is an order in the groups (such as mild, moderate and severe disease), or in the dose of a drug (such as 5 mg/mL, 10 mg/mL, 20 mg/mL) given to the same group of patients, then a linear trend estimation should be used. | [
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Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case can be covered by a t-test. When there are only two means to compare, the t-test and the ANOVA F-test are equivalent; the relation between ANOVA and t is given by . Factorial ANOVA is used when there is more than one factor. Repeated measures ANOVA is used when the same subjects are used for each factor (e.g., in a longitudinal study). Multivariate analysis of variance (MANOVA) is used when there is more than one response variable. Cautions
Balanced experiments (those with an equal sample size for each treatment) are relatively easy to interpret; unbalanced experiments offer more complexity. For single-factor (one-way) ANOVA, the adjustment for unbalanced data is easy, but the unbalanced analysis lacks both robustness and power. For more complex designs the lack of balance leads to further complications. "The orthogonality property of main effects and interactions present in balanced data does not carry over to the unbalanced case. This means that the usual analysis of variance techniques do not apply. Consequently, the analysis of unbalanced factorials is much more difficult than that for balanced designs." In the general case, "The analysis of variance can also be applied to unbalanced data, but then the sums of squares, mean squares, and F-ratios will depend on the order in which the sources of variation are considered." ANOVA is (in part) a test of statistical significance. | [
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The American Psychological Association (and many other organisations) holds the view that simply reporting statistical significance is insufficient and that reporting confidence bounds is preferred. Generalizations
ANOVA is considered to be a special case of linear regression which in turn is a special case of the general linear model. All consider the observations to be the sum of a model (fit) and a residual (error) to be minimized. The Kruskal–Wallis test and the Friedman test are nonparametric tests, which do not rely on an assumption of normality. Connection to linear regression
Below we make clear the connection between multi-way ANOVA and linear regression. Linearly re-order the data so that -th observation is associated with a response and factors where denotes the different factors and is the total number of factors. In one-way ANOVA and in two-way ANOVA . Furthermore, we assume the -th factor has levels, namely . Now, we can one-hot encode the factors into the dimensional vector . The one-hot encoding function is defined such that the -th entry of is
The vector is the concatenation of all of the above vectors for all . Thus, . In order to obtain a fully general -way interaction ANOVA we must also concatenate every additional interaction term in the vector and then add an intercept term. Let that vector be . With this notation in place, we now have the exact connection with linear regression. We simply regress response against the vector . However, there is a concern about identifiability. In order to overcome such issues we assume that the sum of the parameters within each set of interactions is equal to zero. | [
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] |
From here, one can use F-statistics or other methods to determine the relevance of the individual factors. Example
We can consider the 2-way interaction example where we assume that the first factor has 2 levels and the second factor has 3 levels. Define if and if , i.e. is the one-hot encoding of the first factor and is the one-hot encoding of the second factor. With that,
where the last term is an intercept term. For a more concrete example suppose that
Then,
See also
ANOVA on ranks
ANOVA-simultaneous component analysis
Analysis of covariance (ANCOVA)
Analysis of molecular variance (AMOVA)
Analysis of rhythmic variance (ANORVA)
Explained variation
Linear trend estimation
Mixed-design analysis of variance
Multivariate analysis of covariance (MANCOVA)
Permutational analysis of variance
Variance decomposition
Expected mean squares
Footnotes
Notes
References
Pre-publication chapters are available on-line. Cohen, Jacob (1988). Statistical power analysis for the behavior sciences (2nd ed.). Routledge
Cox, David R. (1958). Planning of experiments. Reprinted as
Freedman, David A.(2005). Statistical Models: Theory and Practice, Cambridge University Press. Lehmann, E.L. (1959) Testing Statistical Hypotheses. John Wiley & Sons. Moore, David S. & McCabe, George P. (2003). Introduction to the Practice of Statistics (4e). W H Freeman & Co.
Rosenbaum, Paul R. (2002). Observational Studies (2nd ed.). New York: Springer-Verlag. Further reading
Cox, David R. & Reid, Nancy M. (2000). The theory of design of experiments. (Chapman & Hall/CRC). | [
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] |
Freedman, David A.; Pisani, Robert; Purves, Roger (2007) Statistics, 4th edition. W.W. Norton & Company
Tabachnick, Barbara G. & Fidell, Linda S. (2007). Using Multivariate Statistics (5th ed.). Boston: Pearson International Edition. External links
SOCR ANOVA Activity
Examples of all ANOVA and ANCOVA models with up to three treatment factors, including randomized block, split plot, repeated measures, and Latin squares, and their analysis in R (University of Southampton)
NIST/SEMATECH e-Handbook of Statistical Methods, section 7.4.3: "Are the means equal?" Analysis of variance: Introduction
Design of experiments
Statistical tests
Parametric statistics | [
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Assistive technology (AT) is a term for assistive, adaptive, and rehabilitative devices for people with disabilities and the elderly. People with disabilities often have difficulty performing activities of daily living (ADLs) independently, or even with assistance. ADLs are self-care activities that include toileting, mobility (ambulation), eating, bathing, dressing, grooming, and personal device care. Assistive technology can ameliorate the effects of disabilities that limit the ability to perform ADLs. Assistive technology promotes greater independence by enabling people to perform tasks they were formerly unable to accomplish, or had great difficulty accomplishing, by providing enhancements to, or changing methods of interacting with, the technology needed to accomplish such tasks. For example, wheelchairs provide independent mobility for those who cannot walk, while assistive eating devices can enable people who cannot feed themselves to do so. Due to assistive technology, people with disability have an opportunity of a more positive and easygoing lifestyle, with an increase in "social participation," "security and control," and a greater chance to "reduce institutional costs without significantly increasing household expenses." In schools, assistive technology can be critical in allowing students with disabilities access the general education curriculum. Students who experience challenges writing or keyboarding, for example, can use voice recognition software instead. Adaptive technology
Adaptive technology and assistive technology are different. Assistive technology is something that is used to help disabled people, while adaptive technology covers items that are specifically designed for disabled people and would seldom be used by a non-disabled person. | [
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In other words, assistive technology is any object or system that helps people with disabilities, while adaptive technology is specifically designed for disabled people. Consequently, adaptive technology is a subset of assistive technology. Adaptive technology often refers specifically to electronic and information technology access. Occupational therapy
Occupational therapy (OT) is a healthcare profession that specializes in maintaining or improving the quality of life for individuals that experience challenges when independently performing life's occupations. According to the Occupational Therapy Practice Framework: Domain and Process (3rd ed. ; AOTA, 2014), occupations include areas related to all basic and instrumental activities of daily living (ADLs), rest and sleep, education, work, play, leisure and social participation. Occupational therapists have the specialized skill of employing assistive technology (AT) in the improvement and maintenance of optimal, functional participation in occupations. The application of AT enables an individual to adapt aspects of the environment, that may otherwise be challenging, to the user in order to optimize functional participation in those occupations. As a result, occupational therapists may educate, recommend, and promote the use of AT to improve the quality of life for their clients. Mobility impairments
Wheelchairs
Wheelchairs are devices that can be manually propelled or electrically propelled, and that include a seating system and are designed to be a substitute for the normal mobility that most people have. Wheelchairs and other mobility devices allow people to perform mobility-related activities of daily living which include feeding, toileting, dressing, grooming, and bathing. | [
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The devices come in a number of variations where they can be propelled either by hand or by motors where the occupant uses electrical controls to manage motors and seating control actuators through a joystick, sip-and-puff control, head switches or other input devices. Often there are handles behind the seat for someone else to do the pushing or input devices for caregivers. Wheelchairs are used by people for whom walking is difficult or impossible due to illness, injury, or disability. People with both sitting and walking disability often need to use a wheelchair or walker. Newer advancements in wheelchair design enable wheelchairs to climb stairs, go off-road or propel using segway technology or additional add-ons like handbikes or power assists. Transfer devices
Patient transfer devices generally allow patients with impaired mobility to be moved by caregivers between beds, wheelchairs, commodes, toilets, chairs, stretchers, shower benches, automobiles, swimming pools, and other patient support systems (i.e., radiology, surgical, or examining tables). The most common devices are transfer benches, stretcher or convertible chairs (for lateral, supine transfer), sit-to-stand lifts (for moving patients from one seated position to another i.e., from wheelchairs to commodes), air bearing inflatable mattresses (for supine transfer i.e., transfer from a gurney to an operating room table), gait belts (or transfer belt) and a slider board (or transfer board), usually used for transfer from a bed to a wheelchair or from a bed to an operating table. | [
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Highly dependent patients who cannot assist their caregiver in moving them often require a patient lift (a floor or ceiling-suspended sling lift) which though invented in 1955 and in common use since the early 1960s is still considered the state-of-the-art transfer device by OSHA and the American Nursing Association. Walkers
A walker or walking frame or Rollator is a tool for disabled people who need additional support to maintain balance or stability while walking. It consists of a frame that is about waist high, approximately twelve inches deep and slightly wider than the user. Walkers are also available in other sizes, such as for children, or for heavy people. Modern walkers are height-adjustable. The front two legs of the walker may or may not have wheels attached depending on the strength and abilities of the person using it. It is also common to see caster wheels or glides on the back legs of a walker with wheels on the front. Prosthesis
A prosthesis, prosthetic, or prosthetic limb is a device that replaces a missing body part. It is part of the field of biomechatronics, the science of using mechanical devices with human muscular, musculoskeletal, and nervous systems to assist or enhance motor control lost by trauma, disease, or defect. Prostheses are typically used to replace parts lost by injury (traumatic) or missing from birth (congenital) or to supplement defective body parts. Inside the body, artificial heart valves are in common use with artificial hearts and lungs seeing less common use but under active technology development. | [
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Other medical devices and aids that can be considered prosthetics include hearing aids, artificial eyes, palatal obturator, gastric bands, and dentures. Prostheses are specifically not orthoses, although given certain circumstances a prosthesis might end up performing some or all of the same functionary benefits as an orthosis. Prostheses are technically the complete finished item. For instance, a C-Leg knee alone is not a prosthesis, but only a prosthetic component. The complete prosthesis would consist of the attachment system to the residual limb — usually a "socket", and all the attachment hardware components all the way down to and including the terminal device. Despite the technical difference, the terms are often used interchangeably. The terms "prosthetic" and "orthotic" are adjectives used to describe devices such as a prosthetic knee. The terms "prosthetics" and "orthotics" are used to describe the respective allied health fields. An Occupational Therapist's role in prosthetics include therapy, training and evaluations. Prosthetic training includes orientation to prosthetics components and terminology, donning and doffing, wearing schedule, and how to care for residual limb and the prosthesis. Exoskeletons
A powered exoskeleton is a wearable mobile machine that is powered by a system of electric motors, pneumatics, levers, hydraulics, or a combination of technologies that allow for limb movement with increased strength and endurance. Its design aims to provide back support, sense the user's motion, and send a signal to motors which manage the gears. | [
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The exoskeleton supports the shoulder, waist and thigh, and assists movement for lifting and holding heavy items, while lowering back stress. Adaptive seating and positioning
People with balance and motor function challenges often need specialized equipment to sit or stand safely and securely. This equipment is frequently specialized for specific settings such as in a classroom or nursing home. Positioning is often important in seating arrangements to ensure that user's body pressure is distributed equally without inhibiting movement in a desired way. Positioning devices have been developed to aid in allowing people to stand and bear weight on their legs without risk of a fall. These standers are generally grouped into two categories based on the position of the occupant. Prone standers distribute the body weight to the front of the individual and usually have a tray in front of them. This makes them good for users who are actively trying to carry out some task. Supine standers distribute the body weight to the back and are good for cases where the user has more limited mobility or is recovering from injury. Visual impairments
Many people with serious visual impairments live independently, using a wide range of tools and techniques. Examples of assistive technology for visually impairment include screen readers, screen magnifiers, Braille embossers, desktop video magnifiers, and voice recorders. Screen readers
Screen readers are used to help the visually impaired to easily access electronic information. These software programs run on a computer in order to convey the displayed information through voice (text-to-speech) or braille (refreshable braille displays) in combination with magnification for low vision users in some cases. | [
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There are a variety of platforms and applications available for a variety of costs with differing feature sets. Some example of screen readers are Apple VoiceOver, Google TalkBack and Microsoft Narrator. This software is provided free of charge on all Apple devices. Apple VoiceOver includes the option to magnify the screen, control the keyboard, and provide verbal descriptions to describe what is happening on the screen. There are thirty languages to select from. It also has the capacity to read aloud file content, as well as web pages, E-mail messages, and word processing files. As mentioned above, screen readers may rely on the assistance of text-to-speech tools. To use the text-to-speech tools, the documents must in an electronic form, that is uploaded as the digital format. However, people usually will use the hard copy documents scanned into the computer, which cannot be recognized by the text-to-speech software. To solve this issue, people always use Optical Character Recognition technology accompanied with text-to-speech software. Braille and braille embossers
Braille is a system of raised dots formed into units called braille cells. A full braille cell is made up of six dots, with two parallel rows of three dots, but other combinations and quantities of dots represent other letters, numbers, punctuation marks, or words. People can then use their fingers to read the code of raised dots. A braille embosser is, simply put, a printer for braille. Instead of a standard printer adding ink onto a page, the braille embosser imprints the raised dots of braille onto a page. | [
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Some braille embossers combine both braille and ink so the documents can be read with either sight or touch. Refreshable braille display
A refreshable braille display or braille terminal is an electro-mechanical device for displaying braille characters, usually by means of round-tipped pins raised through holes in a flat surface. Computer users who cannot use a computer monitor use it to read a braille output version of the displayed text. Desktop video magnifier
Desktop video magnifiers are electronic devices that use a camera and a display screen to perform digital magnification of printed materials. They enlarge printed pages for those with low vision. A camera connects to a monitor that displays real-time images, and the user can control settings such as magnification, focus, contrast, underlining, highlighting, and other screen preferences. They come in a variety of sizes and styles; some are small and portable with handheld cameras, while others are much larger and mounted on a fixed stand. Screen magnification software
A screen magnifier is software that interfaces with a computer's graphical output to present enlarged screen content. It allows users to enlarge the texts and graphics on their computer screens for easier viewing. Similar to desktop video magnifiers, this technology assists people with low vision. After the user loads the software into their computer's memory, it serves as a kind of "computer magnifying glass." Wherever the computer cursor moves, it enlarges the area around it. This allows greater computer accessibility for a wide range of visual abilities. Large-print and tactile keyboards
A large-print keyboard has large letters printed on the keys. | [
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On the keyboard shown, the round buttons at the top control software which can magnify the screen (zoom in), change the background color of the screen, or make the mouse cursor on the screen larger. The "bump dots" on the keys, installed in this case by the organization using the keyboards, help the user find the right keys in a tactile way. Navigation assistance
Assistive technology for navigation has exploded on the IEEE Xplore database since 2000, with over 7,500 engineering articles written on assistive technologies and visual impairment in the past 25 years, and over 1,300 articles on solving the problem of navigation for people who are blind or visually impaired. As well, over 600 articles on augmented reality and visual impairment have appeared in the engineering literature since 2000. Most of these articles were published within the past 5 years, and the number of articles in this area is increasing every year. GPS, accelerometers, gyroscopes, and cameras can pinpoint the exact location of the user and provide information on what is in the immediate vicinity, and assistance in getting to a destination. Wearable technology
Wearable technology are smart electronic devices that can be worn on the body as an implant or an accessory. New technologies are exploring how the visually impaired can receive visual information through wearable devices. | [
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Some wearable devices for visual impairment include:
OrCam device
eSight
Brainport
Personal emergency response systems
Personal emergency response systems (PERS), or Telecare (UK term), are a particular sort of assistive technology that use electronic sensors connected to an alarm system to help caregivers manage risk and help vulnerable people stay independent at home longer. An example would be the systems being put in place for senior people such as fall detectors, thermometers (for hypothermia risk), flooding and unlit gas sensors (for people with mild dementia). Notably, these alerts can be customized to the particular person's risks. When the alert is triggered, a message is sent to a caregiver or contact center who can respond appropriately. Accessibility software
In human–computer interaction, computer accessibility (also known as accessible computing) refers to the accessibility of a computer system to all people, regardless of disability or severity of impairment, examples include web accessibility guidelines. Another approach is for the user to present a token to the computer terminal, such as a smart card, that has configuration information to adjust the computer speed, text size, etc. to their particular needs. This is useful where users want to access public computer based terminals in Libraries, ATM, Information kiosks etc. The concept is encompassed by the CEN EN 1332-4 Identification Card Systems – Man-Machine Interface. This development of this standard has been supported in Europe by SNAPI and has been successfully incorporated into the Lasseo specifications, but with limited success due to the lack of interest from public computer terminal suppliers. | [
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Hearing impairments
People in the d/Deaf and hard of hearing community have a more difficult time receiving auditory information as compared to hearing individuals. These individuals often rely on visual and tactile mediums for receiving and communicating information. The use of assistive technology and devices provides this community with various solutions to auditory communication needs by providing higher sound (for those who are hard of hearing), tactile feedback, visual cues and improved technology access. Individuals who are deaf or hard of hearing utilize a variety of assistive technologies that provide them with different access to information in numerous environments. Most devices either provide amplified sound or alternate ways to access information through vision and/or vibration. These technologies can be grouped into three general categories: Hearing Technology, alerting devices, and communication support. Hearing aids
A hearing aid or deaf aid is an electro-acoustic device which is designed to amplify sound for the wearer, usually with the aim of making speech more intelligible, and to correct impaired hearing as measured by audiometry. This type of assistive technology helps people with hearing loss participate more fully in their hearing communities by allowing them to hear more clearly. They amplify any and all sound waves through use of a microphone, amplifier, and speaker. There is a wide variety of hearing aids available, including digital, in-the-ear, in-the-canal, behind-the-ear, and on-the-body aids. Assistive listening devices
Assistive listening devices include FM, infrared, and loop assistive listening devices. | [
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This type of technology allows people with hearing difficulties to focus on a speaker or subject by getting rid of extra background noises and distractions, making places like auditoriums, classrooms, and meetings much easier to participate in. The assistive listening device usually uses a microphone to capture an audio source near to its origin and broadcast it wirelessly over an FM (Frequency Modulation) transmission, IR (Infra Red) transmission, IL (Induction Loop) transmission, or other transmission methods. The person who is listening may use an FM/IR/IL Receiver to tune into the signal and listen at his/her preferred volume. Amplified telephone equipment
This type of assistive technology allows users to amplify the volume and clarity of their phone calls so that they can easily partake in this medium of communication. There are also options to adjust the frequency and tone of a call to suit their individual hearing needs. Additionally, there is a wide variety of amplified telephones to choose from, with different degrees of amplification. For example, a phone with 26 to 40 decibel is generally sufficient for mild hearing loss, while a phone with 71 to 90 decibel is better for more severe hearing loss. Augmentative and alternative communication
Augmentative and alternative communication (AAC) is an umbrella term that encompasses methods of communication for those with impairments or restrictions on the production or comprehension of spoken or written language. AAC systems are extremely diverse and depend on the capabilities of the user. | [
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They may be as basic as pictures on a board that are used to request food, drink, or other care; or they can be advanced speech generating devices, based on speech synthesis, that are capable of storing hundreds of phrases and words. Cognitive impairments
Assistive Technology for Cognition (ATC) is the use of technology (usually high tech) to augment and assist cognitive processes such as attention, memory, self-regulation, navigation, emotion recognition and management, planning, and sequencing activity. Systematic reviews of the field have found that the number of ATC are growing rapidly, but have focused on memory and planning, that there is emerging evidence for efficacy, that a lot of scope exists to develop new ATC. Examples of ATC include: NeuroPage which prompts users about meetings, Wakamaru, which provides companionship and reminds users to take medicine and calls for help if something is wrong, and telephone Reassurance systems. Memory aids
Memory aids are any type of assistive technology that helps a user learn and remember certain information. Many memory aids are used for cognitive impairments such as reading, writing, or organizational difficulties. For example, a Smartpen records handwritten notes by creating both a digital copy and an audio recording of the text. Users simply tap certain parts of their notes, the pen saves it, and reads it back to them. From there, the user can also download their notes onto a computer for increased accessibility. Digital voice recorders are also used to record "in the moment" information for fast and easy recall at a later time. | [
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Educational software
Educational software is software that assists people with reading, learning, comprehension, and organizational difficulties. Any accommodation software such as text readers, notetakers, text enlargers, organization tools, word predictions, and talking word processors falls under the category of educational software. Eating impairments
Adaptive eating devices include items commonly used by the general population like spoons and forks and plates. However they become assistive technology when they are modified to accommodate the needs of people who have difficulty using standard cutlery due to a disabling condition. Common modifications include increasing the size of the utensil handle to make it easier to grasp. Plates and bowls may have a guard on the edge that stops food being pushed off of the dish when it is being scooped. More sophisticated equipment for eating includes manual and powered feeding devices. These devices support those who have little or no hand and arm function and enable them to eat independently. In sports
Assistive technology in sports is an area of technology design that is growing. Assistive technology is the array of new devices created to enable sports enthusiasts who have disabilities to play. Assistive technology may be used in adaptive sports, where an existing sport is modified to enable players with a disability to participate; or, assistive technology may be used to invent completely new sports with athletes with disabilities exclusively in mind. An increasing number of people with disabilities are participating in sports, leading to the development of new assistive technology. Assistive technology devices can be simple, or "low-technology", or they may use highly advanced technology. | [
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"Low-tech" devices can include velcro gloves and adaptive bands and tubes. "High-tech" devices can include all-terrain wheelchairs and adaptive bicycles. Accordingly, assistive technology can be found in sports ranging from local community recreation to the elite Paralympic Games. More complex assistive technology devices have been developed over time, and as a result, sports for people with disabilities "have changed from being a clinical therapeutic tool to an increasingly competition-oriented activity". In education
In the United States there are two major pieces of legislation that govern the use of assistive technology within the school system. The first is Section 504 of the Rehabilitation Act of 1973 and the second being the Individuals with Disabilities Education Act (IDEA) which was first enacted in 1975 under the name The Education for All Handicapped Children Act. In 2004, during the reauthorization period for IDEA, the National Instructional Material Access Center (NIMAC) was created which provided a repository of accessible text including publisher's textbooks to students with a qualifying disability. Files provided are in XML format and used as a starting platform for braille readers, screen readers, and other digital text software. IDEA defines assistive technology as follows: "any item, piece of equipment, or product system, whether acquired commercially off the shelf, modified, or customized, that is used to increase, maintain, or improve functional capabilities of a child with a disability. (B) Exception.--The term does not include a medical device that is surgically implanted, or the replacement of such device." | [
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Assistive technology listed is a student's IEP is not only recommended, it is required (Koch, 2017). These devices help students both with and without disabilities access the curriculum in a way they were previously unable to (Koch, 2017). Occupational therapists play an important role in educating students, parents and teachers about the assistive technology they may interact with (Koch, 2017). Assistive technology in this area is broken down into low, mid, and high tech categories. Low tech encompasses equipment that is often low cost and does not include batteries or requires charging. Examples include adapted paper and pencil grips for writing or masks and color overlays for reading. Mid tech supports used in the school setting include the use of handheld spelling dictionaries and portable word processors used to keyboard writing. High tech supports involve the use of tablet devices and computers with accompanying software. Software supports for writing include the use of auditory feedback while keyboarding, word prediction for spelling, and speech to text. Supports for reading include the use of text to speech (TTS) software and font modification via access to digital text. Limited supports are available for math instruction and mostly consist of grid based software to allow younger students to keyboard equations and auditory feedback of more complex equations using MathML and Daisy. Dementia care
Assistive technology for memory support
A 2017 Cochrane Review highlighted the current lack of high-quality evidence to determine whether assistive technology effectively supports people with dementia to manage memory issues. Thus, it is not presently sure whether or not assistive technology is beneficial for memory problems. | [
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Computer accessibility
One of the largest problems that affect disabled people is discomfort with prostheses. An experiment performed in Massachusetts utilized 20 people with various sensors attached to their arms. The subjects tried different arm exercises, and the sensors recorded their movements. All of the data helped engineers develop new engineering concepts for prosthetics. Assistive technology may attempt to improve the ergonomics of the devices themselves such as Dvorak and other alternative keyboard layouts, which offer more ergonomic layouts of the keys. Assistive technology devices have been created to enable disabled people to use modern touch screen mobile computers such as the iPad, iPhone and iPod touch. The Pererro is a plug and play adapter for iOS devices which uses the built in Apple VoiceOver feature in combination with a basic switch. This brings touch screen technology to those who were previously unable to use it. Apple, with the release of iOS 7 had introduced the ability to navigate apps using switch control. Switch access could be activated either through an external bluetooth connected switch, single touch of the screen, or use of right and left head turns using the device's camera. Additional accessibility features include the use of Assistive Touch which allows a user to access multi-touch gestures through pre-programmed onscreen buttons. For users with physical disabilities a large variety of switches are available and customizable to the user's needs varying in size, shape, or amount of pressure required for activation. Switch access may be placed near any area of the body which has consistent and reliable mobility and less subject to fatigue. Common sites include the hands, head, and feet. | [
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Eye gaze and head mouse systems can also be used as an alternative mouse navigation. A user may utilize single or multiple switch sites and the process often involves a scanning through items on a screen and activating the switch once the desired object is highlighted. Home automation
The form of home automation called assistive domotics focuses on making it possible for elderly and disabled people to live independently. Home automation is becoming a viable option for the elderly and disabled who would prefer to stay in their own homes rather than move to a healthcare facility. This field uses much of the same technology and equipment as home automation for security, entertainment, and energy conservation but tailors it towards elderly and disabled users. For example, automated prompts and reminders utilize motion sensors and pre-recorded audio messages; an automated prompt in the kitchen may remind the resident to turn off the oven, and one by the front door may remind the resident to lock the door. Impacts
Overall, assistive technology aims to allow disabled people to "participate more fully in all aspects of life (home, school, and community)" and increases their opportunities for "education, social interactions, and potential for meaningful employment". It creates greater independence and control for disabled individuals. For example, in one study of 1,342 infants, toddlers and preschoolers, all with some kind of developmental, physical, sensory, or cognitive disability, the use of assistive technology created improvements in child development. These included improvements in "cognitive, social, communication, literacy, motor, adaptive, and increases in engagement in learning activities". | [
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] |
Additionally, it has been found to lighten caregiver load. Both family and professional caregivers benefit from assistive technology. Through its use, the time that a family member or friend would need to care for a patient significantly decreases. However, studies show that care time for a professional caregiver increases when assistive technology is used. Nonetheless, their work load is significantly easier as the assistive technology frees them of having to perform certain tasks. There are several platforms that use machine learning to identify the appropriate assistive device to suggest to patients, making assistive devices more accessible. See also
Accessibility
Assisted Living
Augmentative and alternative communication
Braille technology
Design for All (in ICT)
Disability Flag
Durable medical equipment
Matching person and technology model
OATS: Open Source Assistive Technology Software
Occupational Therapy
Transgenerational design
Universal access to education
References
Bibliography
Educational technology
Web accessibility | [
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The abacus (plural abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hindu-Arabic numeral system. The exact origin of the abacus has not yet emerged. It consists of rows of movable beads, or similar objects, strung on a wire. They represent digits. One of the two numbers is set up, and the beads are manipulated to perform an operation such as addition, or even a square or cubic root. In their earliest designs, the rows of beads could be loose on a flat surface or sliding in grooves. Later the beads were made to slide on rods and built into a frame, allowing faster manipulation. Abacuses are still made, often as a bamboo frame with beads sliding on wires. In the ancient world, particularly before the introduction of positional notation, abacuses were a practical calculating tool. The abacus is still used to teach the fundamentals of mathematics to some children, e.g., in post-Soviet states. Designs such as the Japanese soroban have been used for practical calculations of up to multi-digit numbers. Any particular abacus design supports multiple methods to perform calculations, including the four basic operations and square and cube roots. Some of these methods work with non-natural numbers (numbers such as and ). Although calculators and computers are commonly used today instead of abacuses, abacuses remain in everyday use in some countries. | [
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Merchants, traders, and clerks in some parts of Eastern Europe, Russia, China, and Africa use abacuses. The abacus remains in common use
as a scoring system in non-electronic table games. Others may use an abacus due to visual impairment that prevents the use of a calculator. Etymology
The word abacus dates to at least AD 1387 when a Middle English work borrowed the word from Latin that described a sandboard abacus. The Latin word is derived from ancient Greek (abax) which means something without a base, and colloquially, any piece of rectangular material. Alternatively, without reference to ancient texts on etymology, it has been suggested that it means "a square tablet strewn with dust", or "drawing-board covered with dust (for the use of mathematics)" (the exact shape of the Latin perhaps reflects the genitive form of the Greek word, (abakos). While the table strewn with dust definition is popular, some argue evidence is insufficient for that conclusion. Greek probably borrowed from a Northwest Semitic language like Phoenician, evidenced by a cognate with the Hebrew word ʾābāq (), or “dust” (in the post-Biblical sense "sand used as a writing surface"). Both abacuses and abaci (soft or hard "c") are used as plurals. The user of an abacus is called an abacist. History
Mesopotamia
The Sumerian abacus appeared between 2700–2300 BC. It held a table of successive columns which delimited the successive orders of magnitude of their sexagesimal (base 60) number system. | [
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Some scholars point to a character in Babylonian cuneiform that may have been derived from a representation of the abacus. It is the belief of Old Babylonian scholars, such as Ettore Carruccio, that Old Babylonians "may have used the abacus for the operations of addition and subtraction; however, this primitive device proved difficult to use for more complex calculations". Egypt
Greek historian Herodotus mentioned the abacus in Ancient Egypt. He wrote that the Egyptians manipulated the pebbles from right to left, opposite in direction to the Greek left-to-right method. Archaeologists have found ancient disks of various sizes that are thought to have been used as counters. However, wall depictions of this instrument are yet to be discovered. Persia
At around 600 BC, Persians first began to use the abacus, during the Achaemenid Empire. Under the Parthian, Sassanian, and Iranian empires, scholars concentrated on exchanging knowledge and inventions with the countries around them – India, China, and the Roman Empire- which is how the abacus may have been exported to other countries. Greece
The earliest archaeological evidence for the use of the Greek abacus dates to the 5th century BC. Demosthenes (384 BC–322 BC) complained that the need to use pebbles for calculations was too difficult. A play by Alexis from the 4th century BC mentions an abacus and pebbles for accounting, and both Diogenes and Polybius use the abacus as a metaphor for human behavior, stating "that men that sometimes stood for more and sometimes for less" like the pebbles on an abacus. | [
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The Greek abacus was a table of wood or marble, pre-set with small counters in wood or metal for mathematical calculations. This Greek abacus saw use in Achaemenid Persia, the Etruscan civilization, Ancient Rome, and the Western Christian world until the French Revolution. A tablet found on the Greek island Salamis in 1846 AD (the Salamis Tablet) dates to 300 BC, making it the oldest counting board discovered so far. It is a slab of white marble in length, wide, and thick, on which are 5 groups of markings. In the tablet's center is a set of 5 parallel lines equally divided by a vertical line, capped with a semicircle at the intersection of the bottom-most horizontal line and the single vertical line. Below these lines is a wide space with a horizontal crack dividing it. Below this crack is another group of eleven parallel lines, again divided into two sections by a line perpendicular to them, but with the semicircle at the top of the intersection; the third, sixth and ninth of these lines are marked with a cross where they intersect with the vertical line. Also from this time frame, the Darius Vase was unearthed in 1851. It was covered with pictures, including a "treasurer" holding a wax tablet in one hand while manipulating counters on a table with the other. China
The earliest known written documentation of the Chinese abacus dates to the 2nd century BC. The Chinese abacus, also known as the suanpan (算盤/算盘, lit. | [
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"calculating tray"), is typically tall and comes in various widths, depending on the operator. It usually has more than seven rods. There are two beads on each rod in the upper deck and five beads each in the bottom one. The beads are usually rounded and made of hardwood. The beads are counted by moving them up or down towards the beam; beads moved toward the beam are counted, while those moved away from it are not. One of the top beads is 5, while one of the bottom beads is 1. Each rod has a number under it, showing the place value. The suanpan can be reset to the starting position instantly by a quick movement along the horizontal axis to spin all the beads away from the horizontal beam at the center. The prototype of the Chinese abacus appeared during the Han Dynasty, and the beads are oval. The Song Dynasty and earlier used the 1:4 type or four-beads abacus similar to the modern abacus including the shape of the beads commonly known as Japanese-style abacus. In the early Ming Dynasty, the abacus began to appear in a 1:5 ratio. The upper deck had one bead and the bottom had five beads. In the late Ming Dynasty, the abacus styles appeared in a 2:5 ratio. The upper deck had two beads, and the bottom had five. Various calculation techniques were devised for Suanpan enabling efficient calculations. Some schools teach students how to use it. | [
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In the long scroll Along the River During the Qingming Festival painted by Zhang Zeduan during the Song dynasty (960–1297), a suanpan is clearly visible beside an account book and doctor's prescriptions on the counter of an apothecary's (Feibao). The similarity of the Roman abacus to the Chinese one suggests that one could have inspired the other, given evidence of a trade relationship between the Roman Empire and China. However, no direct connection has been demonstrated, and the similarity of the abacuses may be coincidental, both ultimately arising from counting with five fingers per hand. Where the Roman model (like most modern Korean and Japanese) has 4 plus 1 bead per decimal place, the standard suanpan has 5 plus 2. Incidentally, this allows use with a hexadecimal numeral system (or any base up to 18) which may have been used for traditional Chinese measures of weight. (Instead of running on wires as in the Chinese, Korean, and Japanese models, the Roman model used grooves, presumably making arithmetic calculations much slower.) Another possible source of the suanpan is Chinese counting rods, which operated with a decimal system but lacked the concept of zero as a placeholder. The zero was probably introduced to the Chinese in the Tang dynasty (618–907) when travel in the Indian Ocean and the Middle East would have provided direct contact with India, allowing them to acquire the concept of zero and the decimal point from Indian merchants and mathematicians. Rome
The normal method of calculation in ancient Rome, as in Greece, was by moving counters on a smooth table. | [
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Originally pebbles (calculi) were used. Later, and in medieval Europe, jetons were manufactured. Marked lines indicated units, fives, tens, etc. as in the Roman numeral system. This system of 'counter casting' continued into the late Roman empire and in medieval Europe and persisted in limited use into the nineteenth century. Due to Pope Sylvester II's reintroduction of the abacus with modifications, it became widely used in Europe again during the 11th century This abacus used beads on wires, unlike the traditional Roman counting boards, which meant the abacus could be used much faster and was more easily moved. Writing in the 1st century BC, Horace refers to the wax abacus, a board covered with a thin layer of black wax on which columns and figures were inscribed using a stylus. One example of archaeological evidence of the Roman abacus, shown nearby in reconstruction, dates to the 1st century AD. It has eight long grooves containing up to five beads in each and eight shorter grooves having either one or no beads in each. The groove marked I indicates units, X tens, and so on up to millions. The beads in the shorter grooves denote fives –five units, five tens, etc., essentially in a bi-quinary coded decimal system, related to the Roman numerals. The short grooves on the right may have been used for marking Roman "ounces" (i.e. fractions). | [
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India
The Abhidharmakośabhāṣya of Vasubandhu (316-396), a Sanskrit work on Buddhist philosophy, says that the second-century CE philosopher Vasumitra said that "placing a wick (Sanskrit vartikā) on the number one (ekāṅka) means it is a one while placing the wick on the number hundred means it is called a hundred, and on the number one thousand means it is a thousand". It is unclear exactly what this arrangement may have been. Around the 5th century, Indian clerks were already finding new ways of recording the contents of the abacus. Hindu texts used the term śūnya (zero) to indicate the empty column on the abacus. Japan
In Japan, the abacus is called soroban (, lit. "counting tray"). It was imported from China in the 14th century. It was probably in use by the working class a century or more before the ruling class adopted it, as the class structure obstructed such changes. The 1:4 abacus, which removes the seldom-used second and fifth bead became popular in the 1940s. Today's Japanese abacus is a 1:4 type, four-bead abacus, introduced from China in the Muromachi era. It adopts the form of the upper deck one bead and the bottom four beads. The top bead on the upper deck was equal to five and the bottom one is similar to the Chinese or Korean abacus, and the decimal number can be expressed, so the abacus is designed as a one:four device. The beads are always in the shape of a diamond. | [
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The quotient division is generally used instead of the division method; at the same time, in order to make the multiplication and division digits consistently use the division multiplication. Later, Japan had a 3:5 abacus called 天三算盤, which is now in the Ize Rongji collection of Shansi Village in Yamagata City. Japan also used a 2:5 type abacus. The four-bead abacus spread, and became common around the world. Improvements to the Japanese abacus arose in various places. In China an aluminium frame plastic bead abacus was used. The file is next to the four beads, and pressing the "clearing" button put the upper bead in the upper position, and the lower bead in the lower position. The abacus is still manufactured in Japan even with the proliferation, practicality, and affordability of pocket electronic calculators. The use of the soroban is still taught in Japanese primary schools as part of mathematics, primarily as an aid to faster mental calculation. Using visual imagery can complete a calculation as quickly as a physical instrument. Korea
The Chinese abacus migrated from China to Korea around 1400 AD. Koreans call it jupan (주판), supan (수판) or jusan (주산). The four-beads abacus (1:4) was introduced during the Goryeo Dynasty. The 5:1 abacus was introduced to Korea from China during the Ming Dynasty. Native America
Some sources mention the use of an abacus called a nepohualtzintzin in ancient Aztec culture. This Mesoamerican abacus used a 5-digit base-20 system. | [
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The word Nepōhualtzintzin comes from Nahuatl, formed by the roots; Ne – personal -; pōhual or pōhualli – the account -; and tzintzin – small similar elements. Its complete meaning was taken as: counting with small similar elements. Its use was taught in the Calmecac to the temalpouhqueh , who were students dedicated to taking the accounts of skies, from childhood. The Nepōhualtzintzin was divided into two main parts separated by a bar or intermediate cord. In the left part were four beads. Beads in the first row have unitary values (1, 2, 3, and 4), and on the right side, three beads had values of 5, 10, and 15, respectively. In order to know the value of the respective beads of the upper rows, it is enough to multiply by 20 (by each row), the value of the corresponding count in the first row. The device featured 13 rows with 7 beads, 91 in total. This was a basic number for this culture. It had a close relation to natural phenomena, the underworld, and the cycles of the heavens. One Nepōhualtzintzin (91) represented the number of days that a season of the year lasts, two Nepōhualtzitzin (182) is the number of days of the corn's cycle, from its sowing to its harvest, three Nepōhualtzintzin (273) is the number of days of a baby's gestation, and four Nepōhualtzintzin (364) completed a cycle and approximated one year. | [
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When translated into modern computer arithmetic, the Nepōhualtzintzin amounted to the rank from 10 to 18 in floating point, which precisely calculated large and small amounts, although round off was not allowed. The rediscovery of the Nepōhualtzintzin was due to the Mexican engineer David Esparza Hidalgo, who in his travels throughout Mexico found diverse engravings and paintings of this instrument and reconstructed several of them in gold, jade, encrustations of shell, etc. Very old Nepōhualtzintzin are attributed to the Olmec culture, and some bracelets of Mayan origin, as well as a diversity of forms and materials in other cultures. Sanchez wrote in Arithmetic in Maya that another base 5, base 4 abacus had been found in the Yucatán Peninsula that also computed calendar data. This was a finger abacus, on one hand, 0, 1, 2, 3, and 4 were used; and on the other hand 0, 1, 2, and 3 were used. Note the use of zero at the beginning and end of the two cycles. The quipu of the Incas was a system of colored knotted cords used to record numerical data, like advanced tally sticks – but not used to perform calculations. Calculations were carried out using a yupana (Quechua for "counting tool"; see figure) which was still in use after the conquest of Peru. The working principle of a yupana is unknown, but in 2001 Italian mathematician De Pasquale proposed an explanation. | [
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By comparing the form of several yupanas, researchers found that calculations were based using the Fibonacci sequence 1, 1, 2, 3, 5 and powers of 10, 20, and 40 as place values for the different fields in the instrument. Using the Fibonacci sequence would keep the number of grains within any one field at a minimum. Russia
The Russian abacus, the schoty (, plural from , counting), usually has a single slanted deck, with ten beads on each wire (except one wire with four beads for quarter-ruble fractions). Older models have another 4-bead wire for quarter-kopeks, which were minted until 1916. The Russian abacus is often used vertically, with each wire running horizontally. The wires are usually bowed upward in the center, to keep the beads pinned to either side. It is cleared when all the beads are moved to the right. During manipulation, beads are moved to the left. For easy viewing, the middle 2 beads on each wire (the 5th and 6th bead) usually are of a different color from the other eight. Likewise, the left bead of the thousands wire (and the million wire, if present) may have a different color. The Russian abacus was in use in shops and markets throughout the former Soviet Union, and its usage was taught in most schools until the 1990s. Even the 1874 invention of mechanical calculator, Odhner arithmometer, had not replaced them in Russia; according to Yakov Perelman. | [
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Some businessmen attempting to import calculators into the Russian Empire were known to leave in despair after watching a skilled abacus operator. Likewise, the mass production of Felix arithmometers since 1924 did not significantly reduce abacus use in the Soviet Union. The Russian abacus began to lose popularity only after the mass production of domestic microcalculators in 1974. The Russian abacus was brought to France around 1820 by mathematician Jean-Victor Poncelet, who had served in Napoleon's army and had been a prisoner of war in Russia. The abacus had fallen out of use in western Europe in the 16th century with the rise of decimal notation and algorismic methods. To Poncelet's French contemporaries, it was something new. Poncelet used it, not for any applied purpose, but as a teaching and demonstration aid. The Turks and the Armenian people used abacuses similar to the Russian schoty. It was named a coulba by the Turks and a choreb by the Armenians. School abacus
Around the world, abacuses have been used in pre-schools and elementary schools as an aid in teaching the numeral system and arithmetic. In Western countries, a bead frame similar to the Russian abacus but with straight wires and a vertical frame is common (see image). The wireframe may be used either with positional notation like other abacuses (thus the 10-wire version may represent numbers up to 9,999,999,999), or each bead may represent one unit (e.g. 74 can be represented by shifting all beads on 7 wires and 4 beads on the 8th wire, so numbers up to 100 may be represented). | [
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In the bead frame shown, the gap between the 5th and 6th wire, corresponding to the color change between the 5th and the 6th bead on each wire, suggests the latter use. Teaching multiplication, e.g. 6 times 7, may be represented by shifting 7 beads on 6 wires. The red-and-white abacus is used in contemporary primary schools for a wide range of number-related lessons. The twenty bead version, referred to by its Dutch name rekenrek ("calculating frame"), is often used, either on a string of beads or on a rigid framework. Feynman vs the abacus
Physicist Richard Feynman was noted for facility in mathematical calculations. He wrote about an encounter in Brazil with a Japanese abacus expert, who challenged him to speed contests between Feynman's pen and paper, and the abacus. The abacus was much faster for addition, somewhat faster for multiplication, but Feynman was faster at division. When the abacus was used for a really difficult challenge, i.e. cube roots, Feynman won easily. However, the number chosen at random was close to a number Feynman happened to know was an exact cube, allowing him to use approximate methods. Neurological analysis
Learning how to calculate with the abacus may improve capacity for mental calculation. Abacus-based mental calculation (AMC), which was derived from the abacus, is the act of performing calculations, including addition, subtraction, multiplication, and division, in the mind by manipulating an imagined abacus. It is a high-level cognitive skill that runs calculations with an effective algorithm. | [
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People doing long-term AMC training show higher numerical memory capacity and experience more effectively connected neural pathways. They are able to retrieve memory to deal with complex processes. AMC involves both visuospatial and visuomotor processing that generate the visual abacus and move the imaginary beads. Since it only requires that the final position of beads be remembered, it takes less memory and less computation time. Renaissance abacuses
Binary abacus
The binary abacus is used to explain how computers manipulate numbers. The abacus shows how numbers, letters, and signs can be stored in a binary system on a computer, or via ASCII. The device consists of a series of beads on parallel wires arranged in three separate rows. The beads represent a switch on the computer in either an "on" or "off" position. Visually impaired users
An adapted abacus, invented by Tim Cranmer, and called a Cranmer abacus is commonly used by visually impaired users. A piece of soft fabric or rubber is placed behind the beads, keeping them in place while the users manipulate them. The device is then used to perform the mathematical functions of multiplication, division, addition, subtraction, square root, and cube root. Although blind students have benefited from talking calculators, the abacus is often taught to these students in early grades. Blind students can also complete mathematical assignments using a braille-writer and Nemeth code (a type of braille code for mathematics) but large multiplication and long division problems are tedious. | [
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The abacus gives these students a tool to compute mathematical problems that equals the speed and mathematical knowledge required by their sighted peers using pencil and paper. Many blind people find this number machine a useful tool throughout life. See also
Chinese Zhusuan
Chisanbop
Logical abacus
Mental abacus
Napier's bones
Sand table
Slide rule
Soroban
Suanpan
Notes
Footnotes
References
Reading
External links
Tutorials
Min Multimedia
Abacus curiosities
Abacus in Various Number Systems at cut-the-knot
Java applet of Chinese, Japanese and Russian abaci
An atomic-scale abacus
Examples of Abaci
Aztex Abacus
Indian Abacus
Mathematical tools
Chinese mathematics
Egyptian mathematics
Greek mathematics
Indian mathematics
Japanese mathematics
Roman mathematics | [
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The American National Standards Institute (ANSI ) is a private non-profit organization that oversees the development of voluntary consensus standards for products, services, processes, systems, and personnel in the United States. The organization also coordinates U.S. standards with international standards so that American products can be used worldwide. ANSI accredits standards that are developed by representatives of other standards organizations, government agencies, consumer groups, companies, and others. These standards ensure that the characteristics and performance of products are consistent, that people use the same definitions and terms, and that products are tested the same way. ANSI also accredits organizations that carry out product or personnel certification in accordance with requirements defined in international standards. The organization's headquarters are in Washington, D.C. ANSI's operations office is located in New York City. The ANSI annual operating budget is funded by the sale of publications, membership dues and fees, accreditation services, fee-based programs, and international standards programs. History
ANSI was most likely originally formed in 1918, when five engineering societies and three government agencies founded the American Engineering Standards Committee (AESC). In 1928, the AESC became the American Standards Association (ASA). In 1966, the ASA was reorganized and became United States of America Standards Institute (USASI). The present name was adopted in 1969. | [
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Prior to 1918, these five founding engineering societies:
American Institute of Electrical Engineers (AIEE, now IEEE)
American Society of Mechanical Engineers (ASME)
American Society of Civil Engineers (ASCE)
American Institute of Mining Engineers (AIME, now American Institute of Mining, Metallurgical, and Petroleum Engineers)
American Society for Testing and Materials (now ASTM International)
had been members of the United Engineering Society (UES). At the behest of the AIEE, they invited the U.S. government Departments of War, Navy (combined in 1947 to become the Department of Defense or DOD) and Commerce to join in founding a national standards organization. According to Adam Stanton, the first permanent secretary and head of staff in 1919, AESC started as an ambitious program and little else. Staff for the first year consisted of one executive, Clifford B. LePage, who was on loan from a founding member, ASME. An annual budget of $7,500 was provided by the founding bodies. In 1931, the organization (renamed ASA in 1928) became affiliated with the U.S. National Committee of the International Electrotechnical Commission (IEC), which had been formed in 1904 to develop electrical and electronics standards. Members
ANSI's members are government agencies, organizations, academic and international bodies, and individuals. In total, the Institute represents the interests of more than 270,000 companies and organizations and 30 million professionals worldwide. Process
Although ANSI itself does not develop standards, the Institute oversees the development and use of standards by accrediting the procedures of standards developing organizations. | [
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ANSI accreditation signifies that the procedures used by standards developing organizations meet the institute's requirements for openness, balance, consensus, and due process. ANSI also designates specific standards as American National Standards, or ANS, when the Institute determines that the standards were developed in an environment that is equitable, accessible and responsive to the requirements of various stakeholders. Voluntary consensus standards quicken the market acceptance of products while making clear how to improve the safety of those products for the protection of consumers. There are approximately 9,500 American National Standards that carry the ANSI designation. The American National Standards process involves:
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In many instances, U.S. standards are taken forward to ISO and IEC, through ANSI or the USNC, where they are adopted in whole or in part as international standards. Adoption of ISO and IEC standards as American standards increased from 0.2% in 1986 to 15.5% in May 2012. Standards panels
The Institute administers nine standards panels:
ANSI Homeland Defense and Security Standardization Collaborative (HDSSC)
ANSI Nanotechnology Standards Panel (ANSI-NSP)
ID Theft Prevention and ID Management Standards Panel (IDSP)
ANSI Energy Efficiency Standardization Coordination Collaborative (EESCC)
Nuclear Energy Standards Coordination Collaborative (NESCC)
Electric Vehicles Standards Panel (EVSP)
ANSI-NAM Network on Chemical Regulation
ANSI Biofuels Standards Coordination Panel
Healthcare Information Technology Standards Panel (HITSP)
Each of the panels works to identify, coordinate, and harmonize voluntary standards relevant to these areas. In 2009, ANSI and the National Institute of Standards and Technology (NIST) formed the Nuclear Energy Standards Coordination Collaborative (NESCC). NESCC is a joint initiative to identify and respond to the current need for standards in the nuclear industry. American national standards
The ASA (as for American Standards Association) photographic exposure system, originally defined in ASA Z38.2.1 (since 1943) and ASA PH2.5 (since 1954), together with the DIN system (DIN 4512 since 1934), became the basis for the ISO system (since 1974), currently used worldwide (ISO 6, ISO 2240, ISO 5800, ISO 12232). A standard for the set of values used to represent characters in digital computers. | [
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The ANSI code standard extended the previously created ASCII seven bit code standard (ASA X3.4-1963), with additional codes for European alphabets (see also Extended Binary Coded Decimal Interchange Code or EBCDIC). In Microsoft Windows, the phrase "ANSI" refers to the Windows ANSI code pages (even though they are not ANSI standards). Most of these are fixed width, though some characters for ideographic languages are variable width. Since these characters are based on a draft of the ISO-8859 series, some of Microsoft's symbols are visually very similar to the ISO symbols, leading many to falsely assume that they are identical. The first computer programming language standard was "American Standard Fortran" (informally known as "FORTRAN 66"), approved in March 1966 and published as ASA X3.9-1966. The programming language COBOL had ANSI standards in 1968, 1974, and 1985. The COBOL 2002 standard was issued by ISO. The original standard implementation of the C programming language was standardized as ANSI X3.159-1989, becoming the well-known ANSI C.
The X3J13 committee was created in 1986 to formalize the ongoing consolidation of Common Lisp, culminating in 1994 with the publication of ANSI's first object-oriented programming standard. A popular Unified Thread Standard for nuts and bolts is ANSI/ASME B1.1 which was defined in 1935, 1949, 1989, and 2003. The ANSI-NSF International standards used for commercial kitchens, such as restaurants, cafeterias, delis, etc. The ANSI/APSP (Association of Pool & Spa Professionals) standards used for pools, spas, hot tubs, barriers, and suction entrapment avoidance. | [
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The ANSI/HI (Hydraulic Institute) standards used for pumps. The ANSI for eye protection is Z87.1, which gives a specific impact resistance rating to the eyewear. This standard is commonly used for shop glasses, shooting glasses, and many other examples of protective eyewear. The ANSI paper sizes (ANSI/ASME Y14.1). Other initiatives
In 2008, ANSI, in partnership with Citation Technologies, created the first dynamic, online web library for ISO 14000 standards. On June 23, 2009, ANSI announced a product and services agreement with Citation Technologies to deliver all ISO Standards on a web-based platform. Through the ANSI-Citation partnership, 17,765 International Standards developed by more than 3,000 ISO technical bodies will be made available on the citation platform, arming subscribers with powerful search tools and collaboration, notification, and change-management functionality. ANSI, in partnership with Citation Technologies, AAMI, ASTM, and DIN, created a single, centralized database for medical device standards on September 9, 2009. In early 2009, ANSI launched a new Certificate Accreditation Program (ANSI-CAP) to provide neutral, third-party attestation that a given certificate program meets the American National Standard ASTM E2659-09. In 2009, ANSI began accepting applications for certification bodies seeking accreditation according to requirements defined under the Toy Safety Certification Program (TSCP) as the official third-party accreditor of TSCP's product certification bodies. In 2006, ANSI launched www.StandardsPortal.org, an online resource for facilitating more open and efficient trade between international markets in the areas of standards, conformity assessment, and technical regulations. | [
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The atomic number or proton number (symbol Z) of a chemical element is the number of protons found in the nucleus of every atom of that element. The atomic number uniquely identifies a chemical element. It is identical to the charge number of the nucleus. In an uncharged atom, the atomic number is also equal to the number of electrons. The sum of the atomic number Z and the number of neutrons N gives the mass number A of an atom. Since protons and neutrons have approximately the same mass (and the mass of the electrons is negligible for many purposes) and the mass defect of nucleon binding is always small compared to the nucleon mass, the atomic mass of any atom, when expressed in unified atomic mass units (making a quantity called the "relative isotopic mass"), is within 1% of the whole number A. Atoms with the same atomic number but different neutron numbers, and hence different mass numbers, are known as isotopes. A little more than three-quarters of naturally occurring elements exist as a mixture of isotopes (see monoisotopic elements), and the average isotopic mass of an isotopic mixture for an element (called the relative atomic mass) in a defined environment on Earth, determines the element's standard atomic weight. Historically, it was these atomic weights of elements (in comparison to hydrogen) that were the quantities measurable by chemists in the 19th century. | [
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The conventional symbol Z comes from the German word 'number', which, before the modern synthesis of ideas from chemistry and physics, merely denoted an element's numerical place in the periodic table, whose order was then approximately, but not completely, consistent with the order of the elements by atomic weights. Only after 1915, with the suggestion and evidence that this Z number was also the nuclear charge and a physical characteristic of atoms, did the word (and its English equivalent atomic number) come into common use in this context. History
The periodic table and a natural number for each element
Loosely speaking, the existence or construction of a periodic table of elements creates an ordering of the elements, and so they can be numbered in order. Dmitri Mendeleev claimed that he arranged his first periodic tables (first published on March 6, 1869) in order of atomic weight ("Atomgewicht"). However, in consideration of the elements' observed chemical properties, he changed the order slightly and placed tellurium (atomic weight 127.6) ahead of iodine (atomic weight 126.9). This placement is consistent with the modern practice of ordering the elements by proton number, Z, but that number was not known or suspected at the time. A simple numbering based on periodic table position was never entirely satisfactory, however. | [
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Besides the case of iodine and tellurium, later several other pairs of elements (such as argon and potassium, cobalt and nickel) were known to have nearly identical or reversed atomic weights, thus requiring their placement in the periodic table to be determined by their chemical properties. However the gradual identification of more and more chemically similar lanthanide elements, whose atomic number was not obvious, led to inconsistency and uncertainty in the periodic numbering of elements at least from lutetium (element 71) onward (hafnium was not known at this time). The Rutherford-Bohr model and van den Broek
In 1911, Ernest Rutherford gave a model of the atom in which a central nucleus held most of the atom's mass and a positive charge which, in units of the electron's charge, was to be approximately equal to half of the atom's atomic weight, expressed in numbers of hydrogen atoms. This central charge would thus be approximately half the atomic weight (though it was almost 25% different from the atomic number of gold , ), the single element from which Rutherford made his guess). Nevertheless, in spite of Rutherford's estimation that gold had a central charge of about 100 (but was element on the periodic table), a month after Rutherford's paper appeared, Antonius van den Broek first formally suggested that the central charge and number of electrons in an atom was exactly equal to its place in the periodic table (also known as element number, atomic number, and symbolized Z). This proved eventually to be the case. | [
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Moseley's 1913 experiment
The experimental position improved dramatically after research by Henry Moseley in 1913. Moseley, after discussions with Bohr who was at the same lab (and who had used Van den Broek's hypothesis in his Bohr model of the atom), decided to test Van den Broek's and Bohr's hypothesis directly, by seeing if spectral lines emitted from excited atoms fitted the Bohr theory's postulation that the frequency of the spectral lines be proportional to the square of Z. To do this, Moseley measured the wavelengths of the innermost photon transitions (K and L lines) produced by the elements from aluminum (Z = 13) to gold (Z = 79) used as a series of movable anodic targets inside an x-ray tube. The square root of the frequency of these photons increased from one target to the next in an arithmetic progression. This led to the conclusion (Moseley's law) that the atomic number does closely correspond (with an offset of one unit for K-lines, in Moseley's work) to the calculated electric charge of the nucleus, i.e. the element number Z. Among other things, Moseley demonstrated that the lanthanide series (from lanthanum to lutetium inclusive) must have 15 members—no fewer and no more—which was far from obvious from known chemistry at that time. Missing elements
After Moseley's death in 1915, the atomic numbers of all known elements from hydrogen to uranium (Z = 92) were examined by his method. | [
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There were seven elements (with Z < 92) which were not found and therefore identified as still undiscovered, corresponding to atomic numbers 43, 61, 72, 75, 85, 87 and 91. From 1918 to 1947, all seven of these missing elements were discovered. By this time, the first four transuranium elements had also been discovered, so that the periodic table was complete with no gaps as far as curium (Z = 96). The proton and the idea of nuclear electrons
In 1915, the reason for nuclear charge being quantized in units of Z, which were now recognized to be the same as the element number, was not understood. An old idea called Prout's hypothesis had postulated that the elements were all made of residues (or "protyles") of the lightest element hydrogen, which in the Bohr-Rutherford model had a single electron and a nuclear charge of one. However, as early as 1907, Rutherford and Thomas Royds had shown that alpha particles, which had a charge of +2, were the nuclei of helium atoms, which had a mass four times that of hydrogen, not two times. If Prout's hypothesis were true, something had to be neutralizing some of the charge of the hydrogen nuclei present in the nuclei of heavier atoms. In 1917, Rutherford succeeded in generating hydrogen nuclei from a nuclear reaction between alpha particles and nitrogen gas, and believed he had proven Prout's law. He called the new heavy nuclear particles protons in 1920 (alternate names being proutons and protyles). | [
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It had been immediately apparent from the work of Moseley that the nuclei of heavy atoms have more than twice as much mass as would be expected from their being made of hydrogen nuclei, and thus there was required a hypothesis for the neutralization of the extra protons presumed present in all heavy nuclei. A helium nucleus was presumed to be composed of four protons plus two "nuclear electrons" (electrons bound inside the nucleus) to cancel two of the charges. At the other end of the periodic table, a nucleus of gold with a mass 197 times that of hydrogen was thought to contain 118 nuclear electrons in the nucleus to give it a residual charge of +79, consistent with its atomic number. The discovery of the neutron makes Z the proton number
All consideration of nuclear electrons ended with James Chadwick's discovery of the neutron in 1932. An atom of gold now was seen as containing 118 neutrons rather than 118 nuclear electrons, and its positive charge now was realized to come entirely from a content of 79 protons. After 1932, therefore, an element's atomic number Z was also realized to be identical to the proton number of its nuclei. Chemical properties
Each element has a specific set of chemical properties as a consequence of the number of electrons present in the neutral atom, which is Z (the atomic number). The configuration of these electrons follows from the principles of quantum mechanics. The number of electrons in each element's electron shells, particularly the outermost valence shell, is the primary factor in determining its chemical bonding behavior. | [
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] |
Hence, it is the atomic number alone that determines the chemical properties of an element; and it is for this reason that an element can be defined as consisting of any mixture of atoms with a given atomic number. New elements
The quest for new elements is usually described using atomic numbers. As of , all elements with atomic numbers 1 to 118 have been observed. Synthesis of new elements is accomplished by bombarding target atoms of heavy elements with ions, such that the sum of the atomic numbers of the target and ion elements equals the atomic number of the element being created. In general, the half-life of a nuclide becomes shorter as atomic number increases, though undiscovered nuclides with certain "magic" numbers of protons and neutrons may have relatively longer half-lives and comprise an island of stability. A hypothetical element composed only of neutrons has also been proposed and would have atomic number 0. See also
Effective atomic number
Mass number
Neutron number
Atomic theory
Chemical element
History of the periodic table
List of elements by atomic number
Prout's hypothesis
References
Chemical properties
Nuclear physics
Atoms
Dimensionless numbers of chemistry
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Affirming the consequent, sometimes called converse error, fallacy of the converse, or confusion of necessity and sufficiency, is a formal fallacy of taking a true conditional statement (e.g., "If the lamp were broken, then the room would be dark"), and invalidly inferring its converse ("The room is dark, so the lamp is broken"), even though the converse may not be true. This arises when a consequent ("the room would be dark") has more than one other possible antecedent (for example, "the lamp is not plugged in" or "the lamp is in working order, but is switched off"). Converse errors are common in everyday thinking and communication and can result from, among other causes, communication issues, misconceptions about logic, and failure to consider other causes. The opposite statement, denying the consequent, is a valid form of argument. Formal description
Affirming the consequent is the action of taking a true statement and invalidly concluding its converse . The name affirming the consequent derives from using the consequent, Q, of , to conclude the antecedent P. This illogic can be summarized formally as or, alternatively, . The root cause of such a logic error is sometimes failure to realize that just because P is a possible condition for Q, P may not be the only condition for Q, i.e. Q may follow from another condition as well. Affirming the consequent can also result from overgeneralizing the experience of many statements having true converses. If P and Q are "equivalent" statements, i.e. | [
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, it is possible to infer P under the condition Q. For example, the statements "It is August 13, so it is my birthday" and "It is my birthday, so it is August 13" are equivalent and both true consequences of the statement "August 13 is my birthday" (an abbreviated form of ). Using one statement to conclude the other is not an example of affirming the consequent, but some people may misapply the approach. Additional examples
Example 1
One way to demonstrate the invalidity of this argument form is with a counterexample with true premises but an obviously false conclusion. For example:
If Bill Gates owns Fort Knox, then Bill Gates is rich. Bill Gates is rich. Therefore, Bill Gates owns Fort Knox. Owning Fort Knox is not the only way to be rich. Any number of other ways to be rich exist. However, one can affirm with certainty that "if someone is not rich" (non-Q), then "this person does not own Fort Knox" (non-P). This is the contrapositive of the first statement, and it must be true if and only if the original statement is true. Example 2
Here is another useful, obviously-fallacious example, but one that does not require familiarity with who Bill Gates is and what Fort Knox is:
If an animal is a dog, then it has four legs. My cat has four legs. Therefore, my cat is a dog. | [
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Here, it is immediately intuitive that any number of other antecedents ("If an animal is a deer...", "If an animal is an elephant...", "If an animal is a moose...", etc.) can give rise to the consequent ("then it has four legs"), and that it is preposterous to suppose that having four legs must imply that the animal is a dog and nothing else. This is useful as a teaching example since most people can immediately recognize that the conclusion reached must be wrong (intuitively, a cat cannot be a dog), and that the method by which it was reached must therefore be fallacious. Example 3
Arguments of the same form can sometimes seem superficially convincing, as in the following example:
If Brian had been thrown off the top of the Eiffel Tower, then he would be dead. Brian is dead. Therefore, Brian was thrown off the top of the Eiffel Tower. Being thrown off the top of the Eiffel Tower is not the only cause of death, since there exist numerous different causes of death. Affirming the consequent is commonly used in rationalization, and thus appears as a coping mechanism in some people. Example 4
In Catch-22, the chaplain is interrogated for supposedly being "Washington Irving"/"Irving Washington", who has been blocking out large portions of soldiers' letters home. The colonel has found such a letter, but with the Chaplain's name signed. "You can read, though, can't you?" the colonel persevered sarcastically. | [
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] |
"The author signed his name." "That's my name there." "Then you wrote it. Q.E.D." P in this case is 'The chaplain signs his own name', and Q 'The chaplain's name is written'. The chaplain's name may be written, but he did not necessarily write it, as the colonel falsely concludes.See also
List of fallacies
Abductive reasoning
Appeal to consequences
Confusion of the inverse
Denying the antecedent
ELIZA effect
Fallacy of the single cause
Fallacy of the undistributed middle
Inference to the best explanation
Modus ponens Modus tollens Post hoc ergo propter hoc''
Necessity and sufficiency
References
Propositional fallacies
Logic articles needing expert attention | [
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Ambiguity is a type of meaning in which a phrase, statement or resolution is not explicitly defined, making several interpretations plausible. A common aspect of ambiguity is uncertainty. It is thus an attribute of any idea or statement whose intended meaning cannot be definitively resolved according to a rule or process with a finite number of steps. (The ambi- part of the term reflects an idea of "two", as in "two meanings".) The concept of ambiguity is generally contrasted with vagueness. In ambiguity, specific and distinct interpretations are permitted (although some may not be immediately obvious), whereas with information that is vague, it is difficult to form any interpretation at the desired level of specificity. Linguistic forms
Lexical ambiguity is contrasted with semantic ambiguity. The former represents a choice between a finite number of known and meaningful context-dependent interpretations. The latter represents a choice between any number of possible interpretations, none of which may have a standard agreed-upon meaning. This form of ambiguity is closely related to vagueness. Linguistic ambiguity can be a problem in law, because the interpretation of written documents and oral agreements is often of paramount importance. Lexical ambiguity
The lexical ambiguity of a word or phrase pertains to its having more than one meaning in the language to which the word belongs. "Meaning" here refers to whatever should be captured by a good dictionary. For instance, the word "bank" has several distinct lexical definitions, including "financial institution" and "edge of a river". Or consider "apothecary". | [
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One could say "I bought herbs from the apothecary". This could mean one actually spoke to the apothecary (pharmacist) or went to the apothecary (pharmacy). The context in which an ambiguous word is used often makes it evident which of the meanings is intended. If, for instance, someone says "I buried $100 in the bank", most people would not think someone used a shovel to dig in the mud. However, some linguistic contexts do not provide sufficient information to disambiguate a used word. Lexical ambiguity can be addressed by algorithmic methods that automatically associate the appropriate meaning with a word in context, a task referred to as word sense disambiguation. The use of multi-defined words requires the author or speaker to clarify their context, and sometimes elaborate on their specific intended meaning (in which case, a less ambiguous term should have been used). The goal of clear concise communication is that the receiver(s) have no misunderstanding about what was meant to be conveyed. An exception to this could include a politician whose "weasel words" and obfuscation are necessary to gain support from multiple constituents with mutually exclusive conflicting desires from their candidate of choice. Ambiguity is a powerful tool of political science. More problematic are words whose senses express closely related concepts. | [
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"Good", for example, can mean "useful" or "functional" (That's a good hammer), "exemplary" (She's a good student), "pleasing" (This is good soup), "moral" (a good person versus the lesson to be learned from a story), "righteous", etc. "I have a good daughter" is not clear about which sense is intended. The various ways to apply prefixes and suffixes can also create ambiguity ("unlockable" can mean "capable of being unlocked" or "impossible to lock"). Semantic and syntactic ambiguity
Semantic ambiguity occurs when a word, phrase or sentence, taken out of context, has more than one interpretation. In "We saw her duck" (example due to Richard Nordquist), the words "her duck" can refer either
to the person's bird (the noun "duck", modified by the possessive pronoun "her"), or
to a motion she made (the verb "duck", the subject of which is the objective pronoun "her", object of the verb "saw"). Syntactic ambiguity arises when a sentence can have two (or more) different meanings because of the structure of the sentence—its syntax. This is often due to a modifying expression, such as a prepositional phrase, the application of which is unclear. | [
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"He ate the cookies on the couch", for example, could mean that he ate those cookies that were on the couch (as opposed to those that were on the table), or it could mean that he was sitting on the couch when he ate the cookies. "To get in, you will need an entrance fee of $10 or your voucher and your drivers' license." This could mean that you need EITHER ten dollars OR BOTH your voucher and your license. Or it could mean that you need your license AND you need EITHER ten dollars OR a voucher. Only rewriting the sentence, or placing appropriate punctuation can resolve a syntactic ambiguity. For the notion of, and theoretic results about, syntactic ambiguity in artificial, formal languages (such as computer programming languages), see Ambiguous grammar. Usually, semantic and syntactic ambiguity go hand in hand. The sentence "We saw her duck" is also syntactically ambiguous. Conversely, a sentence like "He ate the cookies on the couch" is also semantically ambiguous. Rarely, but occasionally, the different parsings of a syntactically ambiguous phrase result in the same meaning. For example, the command "Cook, cook!" can be parsed as "Cook (noun used as vocative), cook (imperative verb form)! ", but also as "Cook (imperative verb form), cook (noun used as vocative)!". | [
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It is more common that a syntactically unambiguous phrase has a semantic ambiguity; for example, the lexical ambiguity in "Your boss is a funny man" is purely semantic, leading to the response "Funny ha-ha or funny peculiar?" Spoken language can contain many more types of ambiguities which are called phonological ambiguities, where there is more than one way to compose a set of sounds into words. For example, "ice cream" and "I scream". Such ambiguity is generally resolved according to the context. A mishearing of such, based on incorrectly resolved ambiguity, is called a mondegreen. Metonymy involves referring to one entity by the name of a different but closely related entity (for example, using "wheels" to refer to a car, or "Wall Street" to refer to the stock exchanges located on that street or even the entire US financial sector). In the modern vocabulary of critical semiotics, metonymy encompasses any potentially ambiguous word substitution that is based on contextual contiguity (located close together), or a function or process that an object performs, such as "sweet ride" to refer to a nice car. Metonym miscommunication is considered a primary mechanism of linguistic humor. Philosophy
Philosophers (and other users of logic) spend a lot of time and effort searching for and removing (or intentionally adding) ambiguity in arguments because it can lead to incorrect conclusions and can be used to deliberately conceal bad arguments. For example, a politician might say, "I oppose taxes which hinder economic growth", an example of a glittering generality. | [
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Some will think they oppose taxes in general because they hinder economic growth. Others may think they oppose only those taxes that they believe will hinder economic growth. In writing, the sentence can be rewritten to reduce possible misinterpretation, either by adding a comma after "taxes" (to convey the first sense) or by changing "which" to "that" (to convey the second sense) or by rewriting it in other ways. The devious politician hopes that each constituent will interpret the statement in the most desirable way, and think the politician supports everyone's opinion. However, the opposite can also be true—an opponent can turn a positive statement into a bad one if the speaker uses ambiguity (intentionally or not). The logical fallacies of amphiboly and equivocation rely heavily on the use of ambiguous words and phrases. In continental philosophy (particularly phenomenology and existentialism), there is much greater tolerance of ambiguity, as it is generally seen as an integral part of the human condition. Martin Heidegger argued that the relation between the subject and object is ambiguous, as is the relation of mind and body, and part and whole. In Heidegger's phenomenology, Dasein is always in a meaningful world, but there is always an underlying background for every instance of signification. Thus, although some things may be certain, they have little to do with Dasein's sense of care and existential anxiety, e.g., in the face of death. | [
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In calling his work Being and Nothingness an "essay in phenomenological ontology" Jean-Paul Sartre follows Heidegger in defining the human essence as ambiguous, or relating fundamentally to such ambiguity. Simone de Beauvoir tries to base an ethics on Heidegger's and Sartre's writings (The Ethics of Ambiguity), where she highlights the need to grapple with ambiguity: "as long as there have been philosophers and they have thought, most of them have tried to mask it... And the ethics which they have proposed to their disciples has always pursued the same goal. It has been a matter of eliminating the ambiguity by making oneself pure inwardness or pure externality, by escaping from the sensible world or being engulfed by it, by yielding to eternity or enclosing oneself in the pure moment." Ethics cannot be based on the authoritative certainty given by mathematics and logic, or prescribed directly from the empirical findings of science. She states: "Since we do not succeed in fleeing it, let us, therefore, try to look the truth in the face. Let us try to assume our fundamental ambiguity. It is in the knowledge of the genuine conditions of our life that we must draw our strength to live and our reason for acting". Other continental philosophers suggest that concepts such as life, nature, and sex are ambiguous. Corey Anton has argued that we cannot be certain what is separate from or unified with something else: language, he asserts, divides what is not, in fact, separate. | [
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Following Ernest Becker, he argues that the desire to 'authoritatively disambiguate' the world and existence has led to numerous ideologies and historical events such as genocide. On this basis, he argues that ethics must focus on 'dialectically integrating opposites' and balancing tension, rather than seeking a priori validation or certainty. Like the existentialists and phenomenologists, he sees the ambiguity of life as the basis of creativity. Literature and rhetoric
In literature and rhetoric, ambiguity can be a useful tool. Groucho Marx's classic joke depends on a grammatical ambiguity for its humor, for example: "Last night I shot an elephant in my pajamas. How he got in my pajamas, I'll never know". Songs and poetry often rely on ambiguous words for artistic effect, as in the song title "Don't It Make My Brown Eyes Blue" (where "blue" can refer to the color, or to sadness). In the narrative, ambiguity can be introduced in several ways: motive, plot, character. F. Scott Fitzgerald uses the latter type of ambiguity with notable effect in his novel The Great Gatsby. Mathematical notation
Mathematical notation, widely used in physics and other sciences, avoids many ambiguities compared to expression in natural language. However, for various reasons, several lexical, syntactic and semantic ambiguities remain. Names of functions
The ambiguity in the style of writing a function should not be confused with a multivalued function, which can (and should) be defined in a deterministic and unambiguous way. Several special functions still do not have established notations. | [
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] |
Usually, the conversion to another notation requires to scale the argument or the resulting value; sometimes, the same name of the function is used, causing confusions. Examples of such underestablished functions:
Sinc function
Elliptic integral of the third kind; translating elliptic integral form MAPLE to Mathematica, one should replace the second argument to its square, see Talk:Elliptic integral#List of notations; dealing with complex values, this may cause problems. Exponential integral
Hermite polynomial
Expressions
Ambiguous expressions often appear in physical and mathematical texts. It is common practice to omit multiplication signs in mathematical expressions. Also, it is common to give the same name to a variable and a function, for example, . Then, if one sees , there is no way to distinguish whether it means multiplied by , or function evaluated at argument equal to . In each case of use of such notations, the reader is supposed to be able to perform the deduction and reveal the true meaning. Creators of algorithmic languages try to avoid ambiguities. Many algorithmic languages (C++ and Fortran) require the character * as symbol of multiplication. The Wolfram Language used in Mathematica allows the user to omit the multiplication symbol, but requires square brackets to indicate the argument of a function; square brackets are not allowed for grouping of expressions. Fortran, in addition, does not allow use of the same name (identifier) for different objects, for example, function and variable; in particular, the expression f=f(x) is qualified as an error. The order of operations may depend on the context. | [
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In most programming languages, the operations of division and multiplication have equal priority and are executed from left to right. Until the last century, many editorials assumed that multiplication is performed first, for example, is interpreted as ; in this case, the insertion of parentheses is required when translating the formulas to an algorithmic language. In addition, it is common to write an argument of a function without parenthesis, which also may lead to ambiguity. In the scientific journal style, one uses roman letters to denote elementary functions, whereas variables are written using italics. For example, in mathematical journals the expression
does not denote the sine function, but the
product of the three variables
,
,
, although in the informal notation of a slide presentation it may stand for . Commas in multi-component subscripts and superscripts are sometimes omitted; this is also potentially ambiguous notation. For example, in the notation , the reader can only infer from the context whether it means a single-index object, taken with the subscript equal to product of variables , and , or it is an indication to a trivalent tensor. Examples of potentially confusing ambiguous mathematical expressions
An expression such as can be understood to mean either or . Often the author's intention can be understood from the context, in cases where only one of the two makes sense, but an ambiguity like this should be avoided, for example by writing or . The expression means in several texts, though it might be thought to mean , since commonly means . | [
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Conversely, might seem to mean , as this exponentiation notation usually denotes function iteration: in general, means . However, for trigonometric and hyperbolic functions, this notation conventionally means exponentiation of the result of function application. The expression can be interpreted as meaning ; however, it is more commonly understood to mean . Notations in quantum optics and quantum mechanics
It is common to define the coherent states in quantum optics with and states with fixed number of photons with . Then, there is an "unwritten rule": the state is coherent if there are more Greek characters than Latin characters in the argument, and photon state if the Latin characters dominate. The ambiguity becomes even worse, if is used for the states with certain value of the coordinate, and means the state with certain value of the momentum, which may be used in books on quantum mechanics. Such ambiguities easily lead to confusions, especially if some normalized adimensional, dimensionless variables are used. Expression may mean a state with single photon, or the coherent state with mean amplitude equal to 1, or state with momentum equal to unity, and so on. The reader is supposed to guess from the context. Ambiguous terms in physics and mathematics
Some physical quantities do not yet have established notations; their value (and sometimes even dimension, as in the case of the Einstein coefficients), depends on the system of notations. Many terms are ambiguous. Each use of an ambiguous term should be preceded by the definition, suitable for a specific case. | [
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Just like Ludwig Wittgenstein states in Tractatus Logico-Philosophicus: "...Only in the context of a proposition has a name meaning." A highly confusing term is gain. For example, the sentence "the gain of a system should be doubled", without context, means close to nothing. It may mean that the ratio of the output voltage of an electric circuit to the input voltage should be doubled. It may mean that the ratio of the output power of an electric or optical circuit to the input power should be doubled. It may mean that the gain of the laser medium should be doubled, for example, doubling the population of the upper laser level in a quasi-two level system (assuming negligible absorption of the ground-state). The term intensity is ambiguous when applied to light. The term can refer to any of irradiance, luminous intensity, radiant intensity, or radiance, depending on the background of the person using the term. Also, confusions may be related with the use of atomic percent as measure of concentration of a dopant, or resolution of an imaging system, as measure of the size of the smallest detail which still can be resolved at the background of statistical noise. See also Accuracy and precision and its talk. The Berry paradox arises as a result of systematic ambiguity in the meaning of terms such as "definable" or "nameable". Terms of this kind give rise to vicious circle fallacies. Other terms with this type of ambiguity are: satisfiable, true, false, function, property, class, relation, cardinal, and ordinal. | [
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Mathematical interpretation of ambiguity
In mathematics and logic, ambiguity can be considered to be an instance of the logical concept of underdetermination—for example, leaves open what the value of X is—while its opposite is a self-contradiction, also called inconsistency, paradoxicalness, or oxymoron, or in mathematics an inconsistent system—such as , which has no solution. Logical ambiguity and self-contradiction is analogous to visual ambiguity and impossible objects, such as the Necker cube and impossible cube, or many of the drawings of M. C. Escher. Constructed language
Some languages have been created with the intention of avoiding ambiguity, especially lexical ambiguity. Lojban and Loglan are two related languages which have been created for this, focusing chiefly on syntactic ambiguity as well. The languages can be both spoken and written. These languages are intended to provide a greater technical precision over big natural languages, although historically, such attempts at language improvement have been criticized. Languages composed from many diverse sources contain much ambiguity and inconsistency. The many exceptions to syntax and semantic rules are time-consuming and difficult to learn. Biology
In structural biology, ambiguity has been recognized as a problem for studying protein conformations. The analysis of a protein three-dimensional structure consists in dividing the macromolecule into subunits called domains. The difficulty of this task arises from the fact that different definitions of what a domain is can be used (e.g. folding autonomy, function, thermodynamic stability, or domain motions), which sometimes results in a single protein having different—yet equally valid—domain assignments. Christianity and Judaism
Christianity and Judaism employ the concept of paradox synonymously with "ambiguity". | [
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Many Christians and Jews endorse Rudolf Otto's description of the sacred as 'mysterium tremendum et fascinans', the awe-inspiring mystery which fascinates humans. The orthodox Catholic writer G. K. Chesterton regularly employed paradox to tease out the meanings in common concepts which he found ambiguous or to reveal meaning often overlooked or forgotten in common phrases. (The title of one of his most famous books, Orthodoxy, itself employing such a paradox.) Music
In music, pieces or sections which confound expectations and may be or are interpreted simultaneously in different ways are ambiguous, such as some polytonality, polymeter, other ambiguous meters or rhythms, and ambiguous phrasing, or (Stein 2005, p.79) any aspect of music. The music of Africa is often purposely ambiguous. To quote Sir Donald Francis Tovey (1935, p.195), "Theorists are apt to vex themselves with vain efforts to remove uncertainty just where it has a high aesthetic value." Visual art
In visual art, certain images are visually ambiguous, such as the Necker cube, which can be interpreted in two ways. Perceptions of such objects remain stable for a time, then may flip, a phenomenon called multistable perception. The opposite of such ambiguous images are impossible objects. Pictures or photographs may also be ambiguous at the semantic level: the visual image is unambiguous, but the meaning and narrative may be ambiguous: is a certain facial expression one of excitement or fear, for instance? Social psychology and the bystander effect
In social psychology, ambiguity is a factor used in determining peoples' responses to various situations. | [
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High levels of ambiguity in an emergency (e.g. an unconscious man lying on a park bench) make witnesses less likely to offer any sort of assistance, due to the fear that they may have misinterpreted the situation and acted unnecessarily. Alternately, non-ambiguous emergencies (e.g. an injured person verbally asking for help) illicit more consistent intervention and assistance. With regard to the bystander effect, studies have shown that emergencies deemed ambiguous trigger the appearance of the classic bystander effect (wherein more witnesses decrease the likelihood of any of them helping) far more than non-ambiguous emergencies. Computer science
In computer science, the SI prefixes kilo-, mega- and giga- were historically used in certain contexts to mean either the first three powers of 1024 (1024, 10242 and 10243) contrary to the metric system in which these units unambiguously mean one thousand, one million, and one billion. This usage is particularly prevalent with electronic memory devices (e.g. DRAM) addressed directly by a binary machine register where a decimal interpretation makes no practical sense. Subsequently, the Ki, Mi, and Gi prefixes were introduced so that binary prefixes could be written explicitly, also rendering k, M, and G unambiguous in texts conforming to the new standard—this led to a new ambiguity in engineering documents lacking outward trace of the binary prefixes (necessarily indicating the new style) as to whether the usage of k, M, and G remains ambiguous (old style) or not (new style). | [
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] |
1 M (where M is ambiguously 1,000,000 or 1,048,576) is less uncertain than the engineering value 1.0e6 (defined to designate the interval 950,000 to 1,050,000). As non-volatile storage devices begin to exceed 1 GB in capacity (where the ambiguity begins to routinely impact the second significant digit), GB and TB almost always mean 109 and 1012 bytes. See also
References
External links
Collection of Ambiguous or Inconsistent/Incomplete Statements
Leaving out ambiguities when writing
Semantics
Mathematical notation
Concepts in epistemology
Barriers to critical thinking
Formal semantics (natural language) | [
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Arthur Schopenhauer ( , ; 22 February 1788 – 21 September 1860) was a German philosopher. He is best known for his 1818 work The World as Will and Representation (expanded in 1844), which characterizes the phenomenal world as the product of a blind noumenal will. Building on the transcendental idealism of Immanuel Kant, Schopenhauer developed an atheistic metaphysical and ethical system that rejected the contemporaneous ideas of German idealism. He was among the first thinkers in Western philosophy to share and affirm significant tenets of Indian philosophy, such as asceticism, denial of the self, and the notion of the world-as-appearance. His work has been described as an exemplary manifestation of philosophical pessimism. Though his work failed to garner substantial attention during his lifetime, Schopenhauer had a posthumous impact across various disciplines, including philosophy, literature, and science. His writing on aesthetics, morality, and psychology have influenced many thinkers and artists. Those who have cited his influence include philosophers Emil Cioran, Friedrich Nietzsche and Ludwig Wittgenstein, scientists Erwin Schrödinger and Albert Einstein, psychoanalysts Sigmund Freud and Carl Jung, writers Leo Tolstoy, Herman Melville, Thomas Mann, Hermann Hesse, Machado de Assis, Jorge Luis Borges, Marcel Proust and Samuel Beckett, and composers Richard Wagner, Johannes Brahms, Arnold Schoenberg and Gustav Mahler. Life
Early life
Arthur Schopenhauer was born on February 22, 1788, in Danzig (then part of the Polish–Lithuanian Commonwealth; present-day Gdańsk, Poland) on Heiligegeistgasse (present day Św. | [
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Ducha 47), the son of Johanna Schopenhauer (née Trosiener; 1766–1838) and Heinrich Floris Schopenhauer (1747–1805), both descendants of wealthy German-Dutch patrician families. Neither of them was very religious; both supported the French Revolution, and were republicans, cosmopolitans and Anglophiles. When Danzig became part of Prussia in 1793, Heinrich moved to Hamburg—a free city with a republican constitution. His firm continued trading in Danzig where most of their extended families remained. Adele, Arthur's only sibling, was born on July 12, 1797. In 1797, Arthur was sent to Le Havre to live with the family of his father's business associate, Grégoire de Blésimaire. He seemed to enjoy his two-year stay there, learning to speak French and fostering a life-long friendship with Jean Anthime Grégoire de Blésimaire. As early as 1799, Arthur started playing the flute. In 1803, he accompanied his parents on a European tour of Holland, Britain, France, Switzerland, Austria and Prussia. Viewed as primarily a pleasure tour, Heinrich used the opportunity to visit some of his business associates abroad. Heinrich offered Arthur a choice: he could stay at home and start preparations for university, or he could travel with them and continue his merchant education. Arthur chose to travel with them. He deeply regretted his choice later because the merchant training was very tedious. He spent twelve weeks of the tour attending school in Wimbledon, where he was disillusioned by strict and intellectually shallow Anglican religiosity. He continued to sharply criticize Anglican religiosity later in life despite his general Anglophilia. | [
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He was also under pressure from his father, who became very critical of his educational results. In 1805, Heinrich drowned in a canal near their home in Hamburg. Although it was possible that his death was accidental, his wife and son believed that it was suicide. He was prone to anxiety and depression; each becoming more pronounced later in his life. Heinrich had become so fussy, even his wife started to doubt his mental health. "There was, in the father's life, some dark and vague source of fear which later made him hurl himself to his death from the attic of his house in Hamburg." Arthur showed similar moodiness during his youth and often acknowledged that he inherited it from his father. There were other instances of serious mental health history on his father's side of the family. Despite his hardship, Schopenhauer liked his father and later referred to him in a positive light. Heinrich Schopenhauer left the family with a significant inheritance that was split in three among Johanna and the children. Arthur Schopenhauer was entitled to control of his part when he reached the age of majority. He invested it conservatively in government bonds and earned annual interest that was more than double the salary of a university professor. After quitting his merchant apprenticeship, with some encouragement from his mother, he dedicated himself to studies at the Ernestine Gymnasium, Gotha, in Saxe-Gotha-Altenburg. While there, he also enjoyed social life among the local nobility, spending large amounts of money, which deeply concerned his frugal mother. | [
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He left the Gymnasium after writing a satirical poem about one of the schoolmasters. Although Arthur claimed that he left voluntarily, his mother's letter indicates that he may have been expelled. Arthur spent two years as a merchant in honor of his dead father. During this time, he had doubts about being able to start a new life as a scholar. Most of his prior education was as a practical merchant and he had trouble learning Latin; a prerequisite for an academic career. His mother moved away, with her daughter Adele, to Weimar—the then centre of German literature—to enjoy social life among writers and artists. Arthur and his mother did not part on good terms. In one letter, she wrote: "You are unbearable and burdensome, and very hard to live with; all your good qualities are overshadowed by your conceit, and made useless to the world simply because you cannot restrain your propensity to pick holes in other people." His mother, Johanna, was generally described as vivacious and sociable. After they split, they did not meet again. She died 24 years later. Some of Arthur's negative opinions about women may be rooted in his troubled relationship with his mother. Arthur moved to Hamburg to live with his friend Jean Anthime, who was also studying to become a merchant. Education
He moved to Weimar but did not live with his mother, who even tried to discourage him from coming by explaining that they would not get along very well. Their relationship deteriorated even further due to their temperamental differences. | [
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He accused his mother of being financially irresponsible, flirtatious and seeking to remarry, which he considered an insult to his father's memory. His mother, while professing her love to him, criticized him sharply for being moody, tactless, and argumentative, and urged him to improve his behavior so that he would not alienate people. Arthur concentrated on his studies, which were now going very well, and he also enjoyed the usual social life such as balls, parties and theater. By that time Johanna's famous salon was well established among local intellectuals and dignitaries, the most celebrated of them being Goethe. Arthur attended her parties, usually when he knew that Goethe would be there—although the famous writer and statesman seemed not even to notice the young and unknown student. It is possible that Goethe kept a distance because Johanna warned him about her son's depressive and combative nature, or because Goethe was then on bad terms with Arthur's language instructor and roommate, Franz Passow. Schopenhauer was also captivated by the beautiful Karoline Jagemann, mistress of Karl August, Grand Duke of Saxe-Weimar-Eisenach, and he wrote to her his only known love poem. Despite his later celebration of asceticism and negative views of sexuality, Schopenhauer occasionally had sexual affairs—usually with women of lower social status, such as servants, actresses, and sometimes even paid prostitutes. In a letter to his friend Anthime he claims that such affairs continued even in his mature age and admits that he had two out-of-wedlock daughters (born in 1819 and 1836), both of whom died in infancy. | [
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In their youthful correspondence Arthur and Anthime were somewhat boastful and competitive about their sexual exploits—but Schopenhauer seemed aware that women usually did not find him very charming or physically attractive, and his desires often remained unfulfilled. He left Weimar to become a student at the University of Göttingen in 1809. There are no written reasons about why Schopenhauer chose that university instead of the then more famous University of Jena, but Göttingen was known as more modern and scientifically oriented, with less attention given to theology. Law or medicine were usual choices for young men of Schopenhauer's status who also needed career and income; he chose medicine due to his scientific interests. Among his notable professors were Bernhard Friedrich Thibaut, Arnold Hermann Ludwig Heeren, Johann Friedrich Blumenbach, Friedrich Stromeyer, Heinrich Adolf Schrader, Johann Tobias Mayer and Konrad Johann Martin Langenbeck. He studied metaphysics, psychology and logic under Gottlob Ernst Schulze, the author of Aenesidemus, who made a strong impression and advised him to concentrate on Plato and Immanuel Kant. He decided to switch from medicine to philosophy around 1810–11 and he left Göttingen, which did not have a strong philosophy program: besides Schulze, the only other philosophy professor was Friedrich Bouterwek, whom Schopenhauer disliked. He did not regret his medicinal and scientific studies; he claimed that they were necessary for a philosopher, and even in Berlin he attended more lectures in sciences than in philosophy. During his days at Göttingen, he spent considerable time studying, but also continued his flute playing and social life. | [
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His friends included Friedrich Gotthilf Osann, Karl Witte, Christian Charles Josias von Bunsen, and William Backhouse Astor Sr. He arrived at the newly founded University of Berlin for the winter semester of 1811–12. At the same time, his mother had just begun her literary career; she published her first book in 1810, a biography of her friend Karl Ludwig Fernow, which was a critical success. Arthur attended lectures by the prominent post-Kantian philosopher Johann Gottlieb Fichte, but quickly found many points of disagreement with his ; he also found Fichte's lectures tedious and hard to understand. He later mentioned Fichte only in critical, negative terms—seeing his philosophy as a lower-quality version of Kant's and considering it useful only because Fichte's poor arguments unintentionally highlighted some failings of Kantianism. He also attended the lectures of the famous Protestant theologian Friedrich Schleiermacher, whom he also quickly came to dislike. His notes and comments on Schleiermacher's lectures show that Schopenhauer was becoming very critical of religion and moving towards atheism. He learned by self-directed reading; besides Plato, Kant and Fichte he also read the works of Schelling, Fries, Jacobi, Bacon, Locke, and much current scientific literature. He attended philological courses by August Böckh and Friedrich August Wolf and continued his naturalistic interests with courses by Martin Heinrich Klaproth, Paul Erman, Johann Elert Bode, Ernst Gottfried Fischer, Johann Horkel, Friedrich Christian Rosenthal and Hinrich Lichtenstein (Lichtenstein was also a friend whom he met at one of his mother's parties in Weimar). | [
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Early work
Schopenhauer left Berlin in a rush in 1813, fearing that the city could be attacked and that he could be pressed into military service as Prussia had just joined the war against France. He returned to Weimar but left after less than a month, disgusted by the fact that his mother was now living with her supposed lover, Georg Friedrich Konrad Ludwig Müller von Gerstenbergk (1778–1838), a civil servant twelve years younger than her; he considered the relationship an act of infidelity to his father's memory. He settled for a while in Rudolstadt, hoping that no army would pass through the small town. He spent his time in solitude, hiking in the mountains and the Thuringian forest and writing his dissertation, On the Fourfold Root of the Principle of Sufficient Reason. He completed his dissertation at about the same time as the French army was defeated at the Battle of Leipzig. He became irritated by the arrival of soldiers in the town and accepted his mother's invitation to visit her in Weimar. She tried to convince him that her relationship with Gerstenbergk was platonic and that she had no intention of remarrying. But Schopenhauer remained suspicious and often came in conflict with Gerstenbergk because he considered him untalented, pretentious, and nationalistic. His mother had just published her second book, Reminiscences of a Journey in the Years 1803, 1804, and 1805, a description of their family tour of Europe, which quickly became a hit. She found his dissertation incomprehensible and said it was unlikely that anyone would ever buy a copy. | [
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In a fit of temper Arthur told her that people would read his work long after the "rubbish" she wrote was totally forgotten. In fact, although they considered her novels of dubious quality, the Brockhaus publishing firm held her in high esteem because they consistently sold well. Hans Brockhaus (1888–1965) later claimed that his predecessors "saw nothing in this manuscript, but wanted to please one of our best-selling authors by publishing her son's work. We published more and more of her son Arthur's work and today nobody remembers Johanna, but her son's works are in steady demand and contribute to Brockhaus'[s] reputation." He kept large portraits of the pair in his office in Leipzig for the edification of his new editors. Also contrary to his mother's prediction, Schopenhauer's dissertation made an impression on Goethe, to whom he sent it as a gift. Although it is doubtful that Goethe agreed with Schopenhauer's philosophical positions, he was impressed by his intellect and extensive scientific education. Their subsequent meetings and correspondence were a great honor to a young philosopher, who was finally acknowledged by his intellectual hero. They mostly discussed Goethe's newly published (and somewhat lukewarmly received) work on color theory. Schopenhauer soon started writing his own treatise on the subject, On Vision and Colors, which in many points differed from his teacher's. Although they remained polite towards each other, their growing theoretical disagreements—and especially Schopenhauer's extreme self-confidence and tactless criticisms—soon made Goethe become distant again and after 1816 their correspondence became less frequent. | [
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Schopenhauer later admitted that he was greatly hurt by this rejection, but he continued to praise Goethe, and considered his color theory a great introduction to his own. Another important experience during his stay in Weimar was his acquaintance with Friedrich Majer—a historian of religion, orientalist and disciple of Herder—who introduced him to Eastern philosophy (see also Indology). Schopenhauer was immediately impressed by the Upanishads (he called them "the production of the highest human wisdom", and believed that they contained superhuman concepts) and the Buddha, and put them on a par with Plato and Kant. He continued his studies by reading the Bhagavad Gita, an amateurish German journal Asiatisches Magazin and Asiatick Researches by the Asiatic Society. Schopenhauer held a profound respect for Indian philosophy; although he loved Hindu texts, he was more interested in Buddhism, which he came to regard as the best religion. His studies on Hindu and Buddhist texts were constrained by the lack of adequate literature, and the latter were mostly restricted to Early Buddhism. He also claimed that he formulated most of his ideas independently, and only later realized the similarities with Buddhism. Schopenhauer read the Latin translation and praised the Upanishads in his main work, The World as Will and Representation (1819), as well as in his Parerga and Paralipomena (1851), and commented,In the whole world there is no study so beneficial and so elevating as that of the Upanishads. It has been the solace of my life, it will be the solace of my death. | [
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As the relationship with his mother fell to a new low, in May 1814 he left Weimar and moved to Dresden. He continued his philosophical studies, enjoyed the cultural life, socialized with intellectuals and engaged in sexual affairs. His friends in Dresden were Johann Gottlob von Quandt, Friedrich Laun, Karl Christian Friedrich Krause and Ludwig Sigismund Ruhl, a young painter who made a romanticized portrait of him in which he improved some of Schopenhauer's unattractive physical features. His criticisms of local artists occasionally caused public quarrels when he ran into them in public. Schopenhauer's main occupation during his stay in Dresden was his seminal philosophical work, The World as Will and Representation, which he started writing in 1814 and finished in 1818. He was recommended to the publisher Friedrich Arnold Brockhaus by Baron Ferdinand von Biedenfeld, an acquaintance of his mother. Although Brockhaus accepted his manuscript, Schopenhauer made a poor impression because of his quarrelsome and fussy attitude, as well as very poor sales of the book after it was published in December 1818. In September 1818, while waiting for his book to be published and conveniently escaping an affair with a maid that caused an unwanted pregnancy, Schopenhauer left Dresden for a year-long vacation in Italy. He visited Venice, Bologna, Florence, Naples and Milan, travelling alone or accompanied by mostly English tourists he met. He spent the winter months in Rome, where he accidentally met his acquaintance Karl Witte and engaged in numerous quarrels with German tourists in the Caffè Greco, among them Johann Friedrich Böhmer, who also mentioned his insulting remarks and unpleasant character. | [
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He enjoyed art, architecture, and ancient ruins, attended plays and operas, and continued his philosophical contemplation and love affairs. One of his affairs supposedly became serious, and for a while he contemplated marriage to a rich Italian noblewoman—but, despite his mentioning this several times, no details are known and it may have been Schopenhauer exaggerating. He corresponded regularly with his sister Adele and became close to her as her relationship with Johanna and Gerstenbergk also deteriorated. She informed him about their financial troubles as the banking house of A. L. Muhl in Danzig—in which her mother invested their whole savings and Arthur a third of his—was near bankruptcy. Arthur offered to share his assets, but his mother refused and became further enraged by his insulting comments. The women managed to receive only thirty percent of their savings while Arthur, using his business knowledge, took a suspicious and aggressive stance towards the banker and eventually received his part in full. The affair additionally worsened the relationships among all three members of the Schopenhauer family. He shortened his stay in Italy because of the trouble with Muhl and returned to Dresden. Disturbed by the financial risk and the lack of responses to his book he decided to take an academic position since it provided him with both income and an opportunity to promote his views. He contacted his friends at universities in Heidelberg, Göttingen and Berlin and found Berlin most attractive. He scheduled his lectures to coincide with those of the famous philosopher G. W. F. Hegel, whom Schopenhauer described as a "clumsy charlatan". | [
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He was especially appalled by Hegel's supposedly poor knowledge of natural sciences and tried to engage him in a quarrel about it already at his test lecture in March 1820. Hegel was also facing political suspicions at the time, when many progressive professors were fired, while Schopenhauer carefully mentioned in his application that he had no interest in politics. Despite their differences and the arrogant request to schedule lectures at the same time as his own, Hegel still voted to accept Schopenhauer to the university. Only five students turned up to Schopenhauer's lectures, and he dropped out of academia. A late essay, "On University Philosophy", expressed his resentment towards the work conducted in academies. Later life
After his tenure in academia, he continued to travel extensively, visiting Leipzig, Nuremberg, Stuttgart, Schaffhausen, Vevey, Milan and spending eight months in Florence. Before he left for his three-year travel, Schopenhauer had an incident with his Berlin neighbor, 47-year-old seamstress Caroline Louise Marquet. The details of the August 1821 incident are unknown. He claimed that he had just pushed her from his entrance after she had rudely refused to leave, and that she had purposely fallen to the ground so that she could sue him. She claimed that he had attacked her so violently that she had become paralyzed on her right side and unable to work. She immediately sued him, and the process lasted until May 1827, when a court found Schopenhauer guilty and forced him to pay her an annual pension until her death in 1842. | [
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Schopenhauer enjoyed Italy, where he studied art and socialized with Italian and English nobles. It was his last visit to the country. He left for Munich and stayed there for a year, mostly recuperating from various health issues, some of them possibly caused by venereal diseases (the treatment his doctor used suggests syphilis). He contacted publishers, offering to translate Hume into German and Kant into English, but his proposals were declined. Returning to Berlin, he began to study Spanish so he could read some of his favorite authors in their original language. He liked Pedro Calderón de la Barca, Lope de Vega, Miguel de Cervantes, and especially Baltasar Gracián. He also made failed attempts to publish his translations of their works. Few attempts to revive his lectures—again scheduled at the same time as Hegel's—also failed, as did his inquiries about relocating to other universities. During his Berlin years, Schopenhauer occasionally mentioned his desire to marry and have a family. For a while he was unsuccessfully courting 17-year-old Flora Weiss, who was 22 years younger than himself. His unpublished writings from that time show that he was already very critical of monogamy but still not advocating polygyny—instead musing about a polyamorous relationship that he called "tetragamy". He had an on-and-off relationship with a young dancer, Caroline Richter (she also used the surname Medon after one of her ex-lovers). They met when he was 33 and she was 19 and working at the Berlin Opera. | [
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She had already had numerous lovers and a son out of wedlock, and later gave birth to another son, this time to an unnamed foreign diplomat (she soon had another pregnancy but the child was stillborn). As Schopenhauer was preparing to escape from Berlin in 1831, due to a cholera epidemic, he offered to take her with him on the condition that she left her young son behind. She refused and he went alone; in his will he left her a significant sum of money, but insisted that it should not be spent in any way on her second son. Schopenhauer claimed that, in his last year in Berlin, he had a prophetic dream that urged him to escape from the city. As he arrived in his new home in Frankfurt, he supposedly had another supernatural experience, an apparition of his dead father and his mother, who was still alive. This experience led him to spend some time investigating paranormal phenomena and magic. He was quite critical of the available studies and claimed that they were mostly ignorant or fraudulent, but he did believe that there are authentic cases of such phenomena and tried to explain them through his metaphysics as manifestations of the will. Upon his arrival in Frankfurt, he experienced a period of depression and declining health. He renewed his correspondence with his mother, and she seemed concerned that he might commit suicide like his father. By now Johanna and Adele were living very modestly. Johanna's writing did not bring her much income, and her popularity was waning. | [
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Their correspondence remained reserved, and Arthur seemed undisturbed by her death in 1838. His relationship with his sister grew closer and he corresponded with her until she died in 1849. In July 1832 Schopenhauer left Frankfurt for Mannheim but returned in July 1833 to remain there for the rest of his life, except for a few short journeys. He lived alone except for a succession of pet poodles named Atman and Butz. In 1836, he published On the Will in Nature. In 1836, he sent his essay "On the Freedom of the Will" to the contest of the Royal Norwegian Society of Sciences and won the prize for the following year. He sent another essay, "On the Basis of Morality", to the Royal Danish Society for Scientific Studies, but did not win the prize despite being the only contestant. The Society was appalled that several distinguished contemporary philosophers were mentioned in a very offensive manner, and claimed that the essay missed the point of the set topic and that the arguments were inadequate. Schopenhauer, who had been very confident that he would win, was enraged by this rejection. He published both essays as The Two Basic Problems of Ethics. The first edition, published in 1841, again failed to draw attention to his philosophy. In the preface to the second edition, in 1860, he was still pouring insults on the Royal Danish Society. Two years later, after some negotiations, he managed to convince his publisher, Brockhaus, to print the second, updated edition of The World as Will and Representation. | [
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That book was again mostly ignored and the few reviews were mixed or negative. Schopenhauer began to attract some followers, mostly outside academia, among practical professionals (several of them were lawyers) who pursued private philosophical studies. He jokingly referred to them as "evangelists" and "apostles". One of the most active early followers was Julius Frauenstädt, who wrote numerous articles promoting Schopenhauer's philosophy. He was also instrumental in finding another publisher after Brockhaus declined to publish Parerga and Paralipomena, believing that it would be another failure. Though Schopenhauer later stopped corresponding with him, claiming that he did not adhere closely enough to his ideas, Frauenstädt continued to promote Schopenhauer's work. They renewed their communication in 1859 and Schopenhauer named him heir for his literary estate. Frauenstädt also became the editor of the first collected works of Schopenhauer. In 1848, Schopenhauer witnessed violent upheaval in Frankfurt after General Hans Adolf Erdmann von Auerswald and Prince Felix Lichnowsky were murdered. He became worried for his own safety and property. Even earlier in life he had had such worries and kept a sword and loaded pistols near his bed to defend himself from thieves. He gave a friendly welcome to Austrian soldiers who wanted to shoot revolutionaries from his window and as they were leaving he gave one of the officers his opera glasses to help him monitor rebels. The rebellion passed without any loss to Schopenhauer and he later praised Alfred I, Prince of Windisch-Grätz for restoring order. | [
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He even modified his will, leaving a large part of his property to a Prussian fund that helped soldiers who became invalids while fighting rebellion in 1848 or the families of soldiers who died in battle. As Young Hegelians were advocating change and progress, Schopenhauer claimed that misery is natural for humans and that, even if some utopian society were established, people would still fight each other out of boredom, or would starve due to overpopulation. In 1851, Schopenhauer published Parerga and Paralipomena, which, as the title says, contains essays that are supplementary to his main work. It was his first successful, widely read book, partly due to the work of his disciples who wrote praising reviews. The essays that proved most popular were the ones that actually did not contain the basic philosophical ideas of his system. Many academic philosophers considered him a great stylist and cultural critic but did not take his philosophy seriously. His early critics liked to point out similarities of his ideas to those Fichte and Schelling, or to claim that there were numerous contradictions in his philosophy. Both criticisms enraged Schopenhauer. He was becoming less interested in intellectual fights, but encouraged his disciples to do so. His private notes and correspondence show that he acknowledged some of the criticisms regarding contradictions, inconsistencies, and vagueness in his philosophy, but claimed that he was not concerned about harmony and agreement in his propositions and that some of his ideas should not be taken literally but instead as metaphors. Academic philosophers were also starting to notice his work. | [
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In 1856, the University of Leipzig sponsored an essay contest about Schopenhauer's philosophy, which was won by Rudolf Seydel's very critical essay. Schopenhauer's friend Jules Lunteschütz made the first of his four portraits of him—which Schopenhauer did not particularly like—which was soon sold to a wealthy landowner, Carl Ferdinand Wiesike, who built a house to display it. Schopenhauer seemed flattered and amused by this, and would claim that it was his first chapel. As his fame increased, copies of paintings and photographs of him were being sold and admirers were visiting the places where he had lived and written his works. People visited Frankfurt's Englischer Hof to observe him dining. Admirers gave him gifts and asked for autographs. He complained that he still felt isolated due to his not very social nature and the fact that many of his good friends had already died from old age. He remained healthy in his own old age, which he attributed to regular walks no matter the weather and always getting enough sleep. He had a great appetite and could read without glasses, but his hearing had been declining since his youth and he developed problems with rheumatism. He remained active and lucid, continued his reading, writing and correspondence until his death. The numerous notes that he made during these years, amongst others on aging, were published posthumously under the title Senilia. In the spring of 1860 his health began to decline, and he experienced shortness of breath and heart palpitations; in September he suffered inflammation of the lungs and, although he was starting to recover, he remained very weak. | [
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The last friend to visit him was Wilhelm Gwinner; according to him, Schopenhauer was concerned that he would not be able to finish his planned additions to Parerga and Paralipomena but was at peace with dying. He died of pulmonary-respiratory failure on 21 September 1860 while sitting at home on his couch. He died at the age of 72 and had a funeral conducted by a Lutheran minister. Philosophy
The world as representation
Schopenhauer saw his philosophy as an extension of Kant's, and used the results of Kantian epistemological investigation (transcendental idealism) as starting point for his own. Kant had argued that the empirical world is merely a complex of appearances whose existence and connection occur only in our mental representations. Schopenhauer did not deny that the external world existed empirically but followed Kant in claiming that our knowledge and experience of the world is always indirect. Schopenhauer reiterates this in the first sentence of his main work: "The world is my representation (Die Welt ist meine Vorstellung)". Everything that there is for cognition (the entire world) exists simply as an object in relation to a subject—a 'representation' to a subject. Everything that belongs to the world is, therefore, 'subject-dependent'. In Book One of The World as Will and Representation, Schopenhauer considers the world from this angle—that is, insofar as it is representation. Theory of perception
In November 1813 Goethe invited Schopenhauer to help him on his Theory of Colours. Although Schopenhauer considered colour theory a minor matter, he accepted the invitation out of admiration for Goethe. | [
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Nevertheless, these investigations led him to his most important discovery in epistemology: finding a demonstration for the a priori nature of causality. Kant openly admitted that it was Hume's skeptical assault on causality that motivated the critical investigations in his Critique of Pure Reason and gave an elaborate proof to show that causality is a priori. After G. E. Schulze had made it plausible that Kant had not disproven Hume's skepticism, it was up to those loyal to Kant's project to prove this important matter. The difference between the approaches of Kant and Schopenhauer was this: Kant simply declared that the empirical content of perception is "given" to us from outside, an expression with which Schopenhauer often expressed his dissatisfaction. He, on the other hand, was occupied with the questions: how do we get this empirical content of perception; how is it possible to comprehend subjective sensations "limited to my skin" as the objective perception of things that lie "outside" of me? Causality is therefore not an empirical concept drawn from objective perceptions, as Hume had maintained; instead, as Kant had said, objective perception presupposes knowledge of causality. By this intellectual operation, comprehending every effect in our sensory organs as having an external cause, the external world arises. With vision, finding the cause is essentially simplified due to light acting in straight lines. | [
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