Charles Sanders Peirce

**Charles Sanders Peirce** ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician, and scientist. He is sometimes known as "the father of pragmatism". He was known for his works in logic, mathematics, philosophy, scientific methodology, and semiotics. Peirce was born in Cambridge, Massachusetts.

Peirce died on April 19, 1914 in Milford, Pennsylvania at the age of 74.

## Mathematics

Peirce's most important work in pure mathematics was in logical and foundational areas. He also worked on linear algebra, matrices, various geometries, topology and Listing numbers, Bell numbers, graphs, the four-color problem, and continuity.

He worked on applied mathematics in economics, engineering, and map projections, and was very active in probability and statistics.

- Discoveries
<div style="float:right;width:8.5em;text-align:center;margin-right:20px;border:solid 1px #bbb"><div style="margin:2px;background-color:#dddddd;font-size:40pt;height:50pt;line-height:100%">↓</div> <div style="font-size:8pt;line-height:150%">The Peirce arrow,
symbol for "(neither) ... **nor** ...", also called the _Quine dagger_</div></div>
Peirce made many important discoveries in formal logic and foundational mathematics. Most of these discoveries were only appreciated long after he died:

In 1860, he suggested a cardinal arithmetic for infinite numbers, years before any work by Georg Cantor (who completed his dissertation in 1867) and without access to Bernard Bolzano's 1851 (posthumous) _Paradoxien des Unendlichen_.

In 1880–1881, he showed how Boolean algebra could be done via a repeated sufficient single binary operation (logical NOR). Which would be independently discovered by Henry M. Sheffer 33 years laters. (See also De Morgan's Laws.)

In 1881, he worked on the axiomatization of natural number arithmetic, a few years before Richard Dedekind and Giuseppe Peano. In the same paper Peirce gave the first purely cardinal definition of a finite set, years before Dedekind. This definition is now called"Dedekind-finite". Amd also gave a formal definition of an infinite set (Dedekind-infinite), as a set(group of items) that can be put into a one-to-one correspondence(matched one to one for each item in both sets) with one of its proper subsets.

In 1885, he differentiated between first-order and second-order quantification. In the same paper he set out what can be read as the first (primitive) axiomatic set theory, anticipating Zermelo by about two decades (Brady 2000, pp. 132–133).

In 1886, he saw that Boolean calculations could be carried out via electrical switches.  50 years before Claude Shannon.
By the later 1890s he made existential graphs, a diagrammatic notation for the predicate calculus. Based on them are John F. Sowa's conceptual graphs and Sun-Joo Shin's diagrammatic reasoning.

- _The New Elements of Mathematics_

Peirce wrote drafts for an introductory textbook, with the working title _The New Elements of Mathematics_, that presented mathematics from an original standpoint. Those drafts and many other of his previously unpublished mathematical manuscripts finally appeared in _The New Elements of Mathematics by Charles S. Peirce_ (1976), edited by mathematician Carolyn Eisele.

- Nature of mathematics

Peirce agreed with Auguste Comte in regarding mathematics as more basic than philosophy and the special sciences (of nature and mind). Peirce classified mathematics into three subareas: (1) mathematics of logic, (2) discrete series, and (3) pseudo-continua (as he called them, including the real numbers) and continua. Influenced by his father Benjamin, Peirce argued that mathematics studies purely hypothetical objects and is not just the science of quantity but is more broadly the science which draws certain conclusions; that mathematics aids logic, not vice versa; and that logic itself is part of philosophy and is the science of drawing conclusions, certain and otherwise.

### Mathematics of logic

<div style="padding:5px;width:40%;border: 1px solid #a2a9b1; float:right;">Mathematical logic and foundations, some noted articles
- "On an Improvement in Boole's Calculus of Logic" (1867)
- "Description of a Notation for the Logic of Relatives" (1870)
- "On the Algebra of Logic" (1880)
- "A Boolian Algebra with One Constant" (1880 MS)
- "On the Logic of Number" (1881)
- "Note B: The Logic of Relatives" (1883)
- "On the Algebra of Logic: A Contribution to the Philosophy of Notation" (1884/1885)
- "The Logic of Relatives" (1897)
- "The Simplest Mathematics" (1902 MS)
- "Prolegomena to an Apology for Pragmaticism" (1906, on existential graphs)
</div>

### Probability and statistics

Peirce held that science gives statistical probabilities, not certainties, and that spontaneity ("absolute chance") is real (see Tychism on his view). Most of his statistical writings promote the frequency interpretation of probability (objective ratios of cases). Many of his writings doubt (and criticize the use of) probability when such models are not based on objective randomization. Though Peirce was largely a frequentist, his possible world semantics introduced the "propensity" theory of probability before Karl Popper. Peirce (sometimes with Joseph Jastrow) looked into the probability judgments of experimental subjects, "perhaps the very first" estimate of subjective probabilities in experimental psychology and (what came to be called) Bayesian statistics.

Peirce came up with modern statistics in "Illustrations of the Logic of Science" (1877–1878) and "A Theory of Probable Inference" (1883). With a repeated measures design, Charles Sanders Peirce and Joseph Jastrow introduced blinded, controlled randomized experiments in 1884 (Hacking 1990:205) (before Ronald A. Fisher). He invented optimal design for experiments on gravity, in which he "corrected the means". He used correlation and smoothing. Peirce extended the work on outliers by Benjamin Peirce, his father.  He introduced the terms "confidence" and "likelihood" (before Jerzy Neyman and Fisher). (See Stephen Stigler's historical books and Ian Hacking 1990. )