Talk:Mathematical induction

message: Today, I happend to read the section Mathematical_induction#Prefix induction, which mainly talks about issues in computational complexity theory. For the latter reason, I think it better belongs to an article about recursion. In mathematical induction, it is pointless to count the "number of applications" of the induction step that is needed to arrive at P(n); there isn't even a notion like "applying the induction step". In analyzing the computational complexity of a recursive program, on the other hand, the number of recursive calls does matter. - Any suggestions on what to do with this section are welcome. - Jochen Burghardt (talk) 10:54, 11 April 2017 (UTC)
thread_title:  Prefix induction 

message: Commenting on this edit and the subsequent revert.
Jochen's edit summary, when reverting, said that the "distinction between mathematical and philosophical induction should be mentioned", with which I agree.  But I think it is already mentioned; indeed, the paragraph starts with an explanation that mathematical induction is not the same as inductive reasoning.  I have to say that 2600:387:3:805:0:0:0:AF's version reads cleaner to me, and the version to which Jochen reverted strikes me as a bit repetitive. --Trovatore (talk) 07:57, 28 September 2017 (UTC)
replies: text: A question appears in the context of the above: What is the status of proof by exhaustion? Isn't it a form of inductive reasoning?--109.166.136.245 (talk) 12:24, 10 August 2019 (UTC)
thread_title:  Logical and mathematical induction 

message: What is the connection between mathematical induction and the inference rule modus ponens? 213.233.84.61 (talk) 11:40, 19 December 2017 (UTC)
thread_title:  Mathematical induction and Modus ponens 

message: Can you provide any references for this paragraph that you added to this article? Jarble ;(talk) 23:43, 27 December 2017 (UTC)
replies: text: Sorry, I cannot find any reference. I learnt the claim from Math Stackexchange. Golopotw (talk) 06:23, 28 December 2017 (UTC)
thread_title:  ZFC set theory 

message: There is a very unclear phrasing in intro at the second sentence which says that this inference scheme ″establish statements for all natural numbers″. What is the explicite link between the truth values of some sentences and natural numbers?) is what needs clarification and expliciteness.--5.2.200.163 (talk) 14:14, 17 January 2018 (UTC)
thread_title:  Clarification for intro - link between the truth value of (some) sentences and natural numbers 

message: I also that the word counter is used in a (sub)section where it mentions several counters instead of only one/1, in standard variant. Could it be that this word counter is what is needed to clarify the aspect mentioned in the previous section of this talk page?--5.2.200.163 (talk) 14:23, 17 January 2018 (UTC)
thread_title:  Counter - word use (in a section but not in the lede) 

message: Another aspect that needs a less implicite phrasing is the connection between math induction to enumerative induction (and modus ponens), namely the status of math induction as infinite enumerative induction and also a complete induction by infiniteness instead of ordinary complete induction where the set of all cases to analyze has a rather small cardinality.--5.2.200.163 (talk) 14:32, 17 January 2018 (UTC)
thread_title:  Connection of this notion to enumerative induction (and case-based reasoning) 

message: Also the mentioning of well-ordered set in the first 2 sentences of intro is not useful for clarity of the notion. This further notion is too technical to be mentioned in the first sentences of the article! Thoughts?--5.2.200.163 (talk) 14:43, 17 January 2018 (UTC)
thread_title:  Too soon word use in intro 

message: As requested on my talk page, here are my reasons for reverting 5.2.200.163's edits of 16 Jan 2018.
* Most commonly, it is used to establish statements   the set of all natural numbers.Mathematical Induction at www.mathisfun.com
:The word "case" refers proof obligations ("base case", "inductive case") in the article and in common literature, including the cited mathisfun page. In 5.2.200.163's version, it refers to natural numbers, or propositions involving natural numbers. This is a likely source of confusion. Another confusion is the appearance of both quantifiers "unlimited" and "all" in the sentence.
* It is sometimes desirable to prove a statement involving two  natural numbers,
:The paragraph is about induction on two or more variables. The latter were somewhat sloppily called "counters" in the text (I'd suggest to change that to "variables" consistently). However, they are not "indexed" by anything.
*In the See also section: 
:Mathematical induction has already been delimitated from inductive reasoning in the lead. There is no need to mention Enumerative induction again. (Both pages are currently under consideration for a merge). 
:I haven't any idea why Modus ponens should be linked here. It is a (un)related to mathematical induction as any other inference rule mlisted e.g. at Propositional calculus#Basic and derived argument forms.
Best regards - Jochen Burghardt (talk) 18:34, 17 January 2018 (UTC)
replies: text: I agree with Jochen Burghardt.  The changes made by 5.2.163 were unclear.Rick Norwood (talk) 19:40, 18 January 2018 (UTC)
replies: text: Please indicate concrete examples of wording to improve clarity about the above mentioned aspects re the delimitation of math ind from inductive reasoning, explaining clearly why math ind is a deductive inference scheme rather than an inductive one, as could appear at first glance. These specifications are a very useful addition to the content of this article.--5.2.200.163 (talk) 13:38, 19 January 2018 (UTC)

text: I never said/implied that proof by exhaustion is a form of mathematical induction, just a finite complete enumerative induction. Unlike proof by exhaustion, mathematical induction is a form of a ′′infinite′′ (number of cases) complete induction. So ″both″ math ind and pbe are examples of ″complete induction″, either finite or infinite. I′ll insert these specifying sentences about the types of complete induction which I hope are very clear.--5.2.200.163 (talk) 17:28, 23 January 2018 (UTC)
replies: text: If you insert information into the article, please supply a reference that says it is pertinent.Rick Norwood (talk) 11:55, 24 January 2018 (UTC)
thread_title:  Revert of 5.2.200.163's edits of 16 Jan 2018 

message: I think that the last edits by a knowledgeable editor introduced a new entry level to the content of this article - and it is not a lower one. While I certainly do not object to adding higher levels in an article,especially later on, I want to ask, if the newly achieved generalizations, already in the lead, are advantageous for the expected readership. Maybe, changing in the lead from the structure of naturals to well-founded relations on a set erects a barrage that degrades the accessibility of the whole article. Purgy (talk) 12:47, 13 March 2018 (UTC)
replies: text: I agree - the new, more general material should be postponed until later in the article. The beginning should focus just on induction for the natural numbers, which is the most important subject for this particular article. — Carl (CBM · talk) 14:16, 13 March 2018 (UTC)
replies: text: I think induction on general wellfounded relations is most naturally treated at transfinite induction.  We could have a brief summary here, with a  link.  Of course it works even for finite relations, but that's a detail — readers who want to get that involved might as well deal with it in the transfinite context, as it is not really any harder. --Trovatore (talk) 02:43, 14 March 2018 (UTC)

text: Well, honestly, ... I think you dumbed it down to almost nothing, depriving it of almost all perspectives. Purgy (talk) 11:20, 16 March 2018 (UTC)
replies: text: What "perspectives" do you mean? The far most important perspective on induction is that it is used to prove facts about the naturals. The lede still says something about structural induction in third paragraph.  — Carl (CBM · talk) 11:47, 16 March 2018 (UTC)
replies: text:  Some of what I subsumed under perspectives was contained in the IP's (CP. Wirth?) edits. I am just not that much after eradicating of all his traces. When I am well hidden, I dare to think about a less petrified WP, math-wise. :| Purgy (talk) 08:24, 17 March 2018 (UTC)
thread_title:  Well-founded relations? 

message: This section has been removed from the article, but is still mentioned in the section on strong induction as if it were still in the article.  That is a problem that often occurs when entire sections are deleted.  If you delete a section, I recommend you do a search to find any references to the deleted section.  Is Theorem of Noetherian Induction important?  I don't know, but either put it back or take out the reference to it. Rick Norwood (talk) 19:03, 16 March 2018 (UTC)
replies: text: Thanks, I hadn't realized that section had been edited so much.  Of course the goal is for everything to work together. I was trying to minimize the amount of direct reverting, which was harder than I thought given the nature of the IP edits that I was looking at.   I don't think "Noetherian induction" is of high prominence, although a separate short section could be written to give it a mention. — Carl (CBM · talk) 22:51, 16 March 2018 (UTC)
thread_title:  Theorem of Noetherian Induction 

message: This example excludes n < 12 because the proposition to be proven fails for n = 11.
However, the proposition is true for n = 8, 9, or 10.
The given proof by induction does not employ the restriction that n > 11. Therefore, the base case can start with 8, with 9 or with 10 and the step case can proceed and yield the same valid proof.
True n = 11 is a  valid counterexample. But this is only pure coincidence/luck that this counterexample is so easy to detect and avoid.
This example raises a more serious issue: 
* How does mathematical induction principle addresses a possible situation in which a counterexample exists for some n that is yet to be discovered? 
* And further, how can it guarantee that no such n exist?  — Preceding unsigned comment added by UriGeva (talk • contribs) 01:26, 16 February 2019 (UTC)
replies: text: Nevertheless, I used this objection to add a remark in the article, stating that there is no similar induction proof for an n smaller than 12. More general, if a formally valid induction proof has been performed, there cannot be a counterexample. - Jochen Burghardt (talk) 08:32, 16 February 2019 (UTC)
thread_title:  Example: forming dollar amounts by coins 

message: Should the proof technique of extending a result inductively to all functions (first by showing the result for indicators and then linear combinations of indicators (simple functions), then to monotone increasing functions, and finally to all negative and positive functions) be included here or in its own article?Aaronjaneiro (talk) 02:47, 4 March 2019 (UTC)
replies: text: Please, add new threads at the bottom, it's WP-standard. (Use tab "New section" in top bar) Purgy (talk) 08:29, 4 March 2019 (UTC)
thread_title:  Measure Theoretic Induction/Standard Machine 

message: What exactly is the significance of n in P(n)? The wording in the article is unclear on the nature of the n variable from a logical perspective. If a property P is considered as a monadic predicate which has an associated universe set of individual constants which bound the object variable x of the predicate P(x), how does n relate to x?--185.53.196.147 (talk) 02:12, 10 August 2019 (UTC)
replies: text: I see these comments re the status of n and I think it can be statele that n is a variable associated to sequences, namely the number of order of the elements of a sequence. Then the properties depending on n, P(n), is actually a sequence of properties P(xn), n as variable  having as set of values the set of natural numbers. Thus P(xn) as an explicite predicate variable has as universe set (or universe of discourse) the set of natural numbers as the set of individual constants for the range of the universal quantifier tacitly present in the usual form P(n) of the property P.--109.166.136.245 (talk) 12:55, 10 August 2019 (UTC)
replies: text: Thanks for reaching out.  Unfortunately, this is outside of my expertise -- the subtleties in the logical foundations of induction are not something I've put much thought into.  I think about induction naively as a consequence of the defining properties of the natural numbers (Peano axiom #9), but I've never really thought about its foundations in logic.  Someone around here will certainly have some insight. Alsosaid1987 (talk) 15:47, 10 August 2019 (UTC)
replies: text: I undid the 10 Aug edits of 109.166.136.245. Imo, neither the question of 185.53.196.147 nor the answer of 109.166.136.245 makes much sense from a formal logic point of view. If there is an issue at all, it shouldn't be discussed in the article's lead section. Section "Formalization" seems more appropriate, and I think it is sufficiently clear about P. - Jochen Burghardt (talk) 08:13, 14 August 2019 (UTC)
thread_title:  Significance of n in P(n) 

message: The article says that the studied property holds for every natural number. Is this mandatory? Couldn't the property P hold only for subsets of natural number like the even or odd naturals or the multiples of 3, or the set of prime numbers...?--185.53.197.197 (talk) 23:39, 13 September 2019 (UTC)
replies: text: Structural induction can be used for this; it just needs a well-ordering on the set under consideration. Induction on natural numbers is (I believe to remeber) its historical precursor, and a special case. - Jochen Burghardt (talk) 07:04, 14 September 2019 (UTC)
replies: text: Well-ordering absolutely required or something else but related? I see that the linked article says something about a recursively defined structure! I see that well-ordering has a requirement of a least element. This brings the question of validity of math induction in sets without least element, but with a recursively defined structure.--185.53.198.164 (talk) 17:12, 14 September 2019 (UTC)
replies: text: I never saw the word "induction" used for the latter. To prove, e.g., that each integer x satsfies some property p(x), an ordinary induction on the absolute value of x might be appropriate; in this case, ℤ is mapped (by abs(.)) to a well-founded set, viz. ℕ. On the other hand, "generalizing" structural induction to the non-well-founded (w.r.t. its usual order) set ℤ by omitting the base case, would allow one to prove a lot of false statements, like "each x∈ℤ is both odd and even": it is well-known that if x is both odd and even, then x+1 is both even and odd, respectively. Also, in a computer-science sense, recursion always requires some base cases (minimal elements of a well-ordered set) in order to avoid endless-recursion. - Jochen Burghardt (talk) 17:43, 14 September 2019 (UTC)
replies: text: I'm not saying that a generalization should have no base cases. For instance when considering the extension of ordinary factorial (which has the property of recursivity) to negative integers the base case could be 0! like in the case of usual factorial or -1 as the additive inverse (a sort of a mirror image) of the number 1. So the factorial has two possible variants of extension in integers: 1) the mirror image of the positive integers factorial (with possibly the same base case as ordinary factorial), 2) the bidirectional factorial as product of the usual factorial of positive integers and negative integers factorial, the two factorials possibly starting from the same base (0!) case but in opposite ways.--185.53.198.164 (talk) 19:38, 14 September 2019 (UTC)

text: Interesting remark about the involvement of absolute value, this seems to be an equivalent formulation to the informal mirror image label. On the page integer I notice that this set has a total order. Could this be sufficient for integers instead of well-ordering, without lack of base case(s)?--185.53.198.164 (talk) 19:46, 14 September 2019 (UTC)

text: These aspects about integers can be discussed in a separate section below.--185.53.198.164 (talk) 19:52, 14 September 2019 (UTC)
replies: text:  Please see WP:NOTFORUM -- the purpose of this page is to discuss the article, not the subject of the article.  --JBL (talk) 21:05, 14 September 2019 (UTC)
replies: text: I have opened the discussion with the intent of subsequent additions/modifications to article.--185.53.198.164 (talk) 21:29, 14 September 2019 (UTC)
replies: text:  In that case please see our policy on original research. --JBL (talk) 23:12, 14 September 2019 (UTC)
replies: text:  Perhaps there are some modifications which are below the originality threshold which would need a citation. There remains to be seen.--185.53.198.164 (talk) 00:46, 15 September 2019 (UTC)
thread_title:  Subsets of natural numbers 

message: Some aspects having occured in the previous section re the induction on the integers (without omitting base case, well-ordering and total order, etc) can be further analyzed here.--185.53.198.164 (talk) 19:56, 14 September 2019 (UTC)
thread_title:  Induction on the integers 

message: The intro definition can include some more maths jargon to refine the notion. Eg, "Mathematical induction is used to infer if a function f(n) is defined on the set of natural numbers as its domain" or something like that. How'd that be? This also can be made a parenthetical statement. It just seems too informal at this point hence this proposition. Akshay Aher (talk) 15:16, 22 September 2019 (UTC)
replies: text: The property P(n) mentioned in the lead could be e.g. "n belongs to the domain of f"; this way, your example is covered by the lead. - Jochen Burghardt (talk) 16:30, 22 September 2019 (UTC)
replies: text: What is the involvement of n as argument of the function(s)  f(n) or P(n) of math induction? Is it a variable or just a dummy variable, a placeholder of some sort? --93.122.249.16 (talk) 17:50, 4 October 2019 (UTC)

text: I see that below an ordered proposition set is mentioned, the ordering of the proposition being done with numbers of order from the ordered set {0, 1, 2, 3,....n}. This seems to indicate that n is just a number of order.--93.122.249.16 (talk) 18:17, 4 October 2019 (UTC)
thread_title:  Suggestion for mathematical terminology in lead 

message: I see that on the Portuguese version of this article there is a statement which could be translated and inserted here. The statement is: Indução matemática é um método de prova matemática usado para demonstrar a verdade de um número infinito de proposições. --2A02:2F07:D5FF:FFFF:0:0:6478:4403 (talk) 14:08, 30 September 2019 (UTC)
replies: text: The Portuguese sentence is too unspecific. Induction (in its basic form) can be used only if the proposition set has the form { p(0), p(1), p(2), p(3), ... }. This is already said in the lead. - Jochen Burghardt (talk) 18:38, 30 September 2019 (UTC)
thread_title:  Suggestion of addition from Portuguese version of this article 

message: The structure of this type of reasoning (having a base/start case and (successive) inductive case(s)) can be noticed that it is is based on the following chain of implications:
:If p(1), then p(2)                   notation         ( p(1) \to p(2) )    
:p(1)                           
:Therefore, p(2)                
...........
...........
...........
:If p(k), then p(k+1)                 notation         ( p(k) \to p(k+1) ) 
:p(k).
:Therefore, p(k+1)
...........
...........
:Therefore, p(n), ...
(Symbolized with logical operator operator notation,
: p(1) \to p(2), 
: p(1) 
: \vdash p(2) 
: p(2) \to p(3), 
: p(2) 
: \vdash p(3) 
............
: p(k) \to p(k+1), 
: p(k) 
: \vdash p(k+1)  
............
: p(n-1) \to p(n), 
: p(n-1) 
:\vdash p(n) ,...)
I think that this aspect re the chain of forward reasoning should be mentioned in article.--37.251.221.82 (talk) 11:32, 2 October 2019 (UTC)
thread_title:  Chain of succesive implications p(1)→p(2)→.....p(k)→p(k+1)→...p(n) 

message: Some aspects from above sections point to the conclusion that  n (in P(n)) as a variable is just a number of order for an ordered proposition set {p(0), p(1), p(2),.....,p(n)}. Is this a valid understanding?--93.122.249.16 (talk) 18:35, 4 October 2019 (UTC)
replies: text: To reply to this and to your above question: yes, I think, the proposition set could be made an ordered set. However, I don't remember to have seen this somewhere in a book, and I can't imagine that it would lead to new insights. Note that, beyond being ordered, all members of the set must be of the form "p of someting", for a fixed property p. - Jochen Burghardt (talk) 05:59, 5 October 2019 (UTC)
replies: text: I see this discution and I think that distinct symbols p(i) and P(x(i)) are needed. p(i) is a proposition from the ordered set {p(0), p(1), p(2), .......} and P is the symbol of the predicate letter from the structure of the ordered set of propositions pi. The ordering of the propositions pi is given by the succesive implication p(i) →p (i+1), mentioned in a section above.--109.166.135.99 (talk) 15:23, 4 January 2020 (UTC)

text: This distinction is important in cases when the numbering of propositions is different than the value of variable depending on the index in the predicate formula P(xi), such as in the examples of subsets of natural numbers like even or odd, multiples of 3, etc.--109.166.135.99 (talk) 17:02, 4 January 2020 (UTC)

text: For instance the sum xn + an is divisible by the sum x + a for every odd n. In this example n takes values in a subset of N, the odd positive integers.--109.166.135.99 (talk) 17:33, 4 January 2020 (UTC)

text: The predicate expression is E(xn) D(ivisible by) (x + a), E(xn) = xn + an and the whole expression E(xn) D (x + a) can be denoted P(xn) or D(xn).--109.166.135.99 (talk) 17:51, 4 January 2020 (UTC)

text: The order number of proposition in this example (re divisibility) can be every value from {0, 1, 2, 3, 4, 5,.....}, but the order number of the predicate formula is from {1, 3, 5, 7, 9, 11, ....2k + 1}.--109.166.135.99 (talk) 18:06, 4 January 2020 (UTC)
thread_title:  Number of order? - {0, 1, 2, 3,... n}  

message: I think that implicational structure/nature of the inductive step should be stated in article.--86.127.33.116 (talk) 12:15, 30 November 2020 (UTC)
thread_title:  The structure of the inductive step as implication/conditional sentence 

message: It has been claimed that the level of this article is only school-level introductory and as a consequence some logical notions appearing in the structure of this reasoning method should not be linked. I think this claim of just introductory level is unfounded or problematic.--86.127.32.116 (talk) 00:41, 1 December 2020 (UTC)
replies: text: Problematic in the sense that the level varies (as is typical for many articles; we want the introductory stuff first and more technical parts later per WP:TECHNICAL), problematic in the sense that some passages (which ones?) are not written in as introductory a level as they should be, or problematic in that you reject the idea that making articles as readable as possible is a good idea? Be specific. —David Eppstein (talk) 01:28, 1 December 2020 (UTC)
replies: text: I agree with the variation of level as typically/usually done for many articles. In this context the questionable/problematic aspect is the exclusion of aspects considered more technical based on self-limitation of the article only to presumed introductory aspects.--86.127.34.116 (talk) 03:00, 1 December 2020 (UTC)

text: As a consequence of the previous paragraph at least a question appears: Are the logical aspects mentioned in a previous section above (and on an user talk page) considered non-introductory or more technical? If yes, what is the frame that considers logical aspects as non-introductory?--86.127.34.116 (talk) 03:08, 1 December 2020 (UTC)

text: (The mentioned user talk is: User_talk:Jochen_Burghardt#Unneeded_links_(at_mathematical_induction)?)--86.124.195.101 (talk) 00:34, 2 December 2020 (UTC)
thread_title:  Level of this article  

message: The common induction does follow from the well ordering principle, because the well ordering assumed is not just any well ordering, but the transitive closure of n < S(n). This entire section, as well as professor Öhman's article, hinges on a misunderstanding of an elementary definition. Tkjanacek (talk) 22:48, 17 April 2021 (UTC)
replies: text: You may be right in that requiring the transitive closure of n < S(n) to be a well-ordering implies the induction axiom. However, Öhman apparently uses a wider understanding of 'well-ordering principle', and infers correct properties from it, so there is no reason why they shouldn't be presented here. Feel free to add your (claimed) equivalence of induction with your narrow understanding of well-ordering, preferrably with a reference or/and a proof. - Jochen Burghardt (talk) 10:24, 18 April 2021 (UTC)
thread_title:  Relationship to well-ordering of the natural numbers 

message: It is argued that mathematical induction is not equivalent to the well-ordering theorem, and that literature is wrong about that. But I only know literature in which it is indicated that complete mathematical induction is equivalent to the well-ordering theorem. The counterexample does not apply for complete mathematical induction, so I think both are equivalent, and that it is all about mathematical induction versus complete mathematical induction.  — Preceding unsigned comment added by 178.84.82.19 (talk) 17:14, 19 December 2021 (UTC)
thread_title:  Mathematical induction versus Well-ordering theorem 

message: This article is wrong when:
The foundation of this comes from the logical implication/inference rules such as 1 --> 0 = 0 that would imply the inductive step was not possible; but, if possible, one must continue by relying on what is explained at https://en.wikipedia.org/wiki/Mathematical_induction#Formalization for the whole reasoning on which mathematical induction is founded.
2. using the terms "deduce", "deduction" and similar instead of "infer", "inference", etc. when meaning arriving to a conclusion by means of logical devices/principles/etc..- Even if they are used as if they were exchangeable in daily conversations, their use must be rigorous here.
*The source used to support the fact that mathematical induction is deductive (https://earlham.edu/~peters/courses/logsys/math-ind.htm) is broken. And, if it were not, I'd still consider it wrong. Proving successive cases IS NOT a form of deduction as each particular case is at the same "level". Only the "general" case stated in the form of hypothesis is the "real" general case and it is not a proven truth like the one used for deductive reasoning.
George Rodney Maruri Game (talk) 17:42, 13 September 2023 (UTC)
replies: text: Your thoughts on this topic constitute original research and should not be pushed into our article. We can only include content describing the consensus of scholarly research on the topic, based on published sources. Unless you can demonstrate that your position is accepted in scholarly research on this topic, it should not be included. —David Eppstein (talk) 18:10, 13 September 2023 (UTC)
replies: text: In the same vein, I request academic support for your claim that mathematical induction is deductive. This article cites a broken link to back this claim. If not, I demand such claim to be removed. I propose contributors vote and present reputable sources to support either claim in the next couple of months. It should not be difficult to find the correct category for this topic. George Rodney Maruri Game (talk) 21:05, 13 September 2023 (UTC)

text: The contested language does actually omit an important point, which is that, while mathematical induction is deductive reasoning, it's not deductive reasoning just from the base case P(0) and the induction step \forall n [P(n)\implies P(n+1)].  It's deductive reasoning from those two things plus (an instance of) the induction axiom.  George, if the intro were to clarify that point, would it address your concerns? --Trovatore (talk) 21:26, 13 September 2023 (UTC)
replies: text: Thanks, Trovatore. I just read some sources and my understanding is similar to yours. There is an inductive step but the "overall reasoning" is based on what is specified at the "Formalization" sub-section which happens to be classified as deductive in Philosophy. It would be useful for people who come here to be told about this to avoid confusion. I myself was confused for the way this article categorized this mathematical proof method as "deductive" without explaining a source that is not broken. There should always be more sources to avoid this kind of problems. I still think that the terms "deduce" and "infer" must not be used the way they are being used here. It will surely cause confusion.George Rodney Maruri Game (talk) 21:44, 13 September 2023 (UTC)
thread_title:  Mathematical induction is related to induction reasoning internally but, as a whole, it is deductive. This article might use some re-phrasing/re-wording.