Wittgenstein/Foundations of Mathematics

Quotes taken from Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939

Cornell University Press

Summary

Lectures I-X

• Lecture I: Wittgenstein introduces his approach to the foundations of mathematics, clarifying that he will not interfere with mathematicians' work or offer new interpretations, but rather address philosophical puzzles arising from everyday language used in mathematics, like "proof" and "number". He emphasizes the importance of understanding the use of mathematical expressions, not just the pictures they conjure, warning against misunderstandings that arise from assimilating expressions with different functions. The goal is not to make mathematical discoveries but to reframe them as inventions, highlighting differences rather than similarities to resolve linguistic confusion.  
• Lecture II: This lecture explores the concept of "understanding" a mathematical symbol or rule, questioning whether it's a mental state or defined by correct application over time. Wittgenstein examines the 'flash of understanding' and the intention to follow rules (like in chess), arguing that understanding isn't guaranteed by a mental picture or formula but is demonstrated through consistent, correct use, which itself relies on shared techniques and agreement. He discusses the ambiguity of "doing the same thing" when applying a rule like y=x^2 to different numbers, suggesting that following a rule involves decisions based on established practice rather than just intuition.  
• Lecture III: Wittgenstein distinguishes between mathematical propositions (timeless rules for symbol use, like 2+2=4) and non-mathematical propositions (statements about the world using those symbols, which are temporal). He questions what constitutes a mathematical proof, like the standard multiplication algorithm, arguing its status as a proof depends on its reliable application and connection to ordinary language concepts like "proof" or "equals". The lecture concludes that calling a pattern a 'proof' depends on its application, and there isn't a single, absolute mathematical fact separate from the physical applications or the specific proof pattern used.  
• Lecture IV: The lecture explores the relationship between mathematical calculi and physical reality, questioning whether a calculation used solely for prediction (like predicting weights or decorating walls) constitutes mathematics or physics. Wittgenstein argues that mathematical propositions like 25×25=625 function as rules, made independent of experience (unlike predictions), which structure how we describe experiences. He further discusses the impossibility of constructing a heptagon, suggesting this mathematical "impossibility" is a decision to exclude the phrase "construction of the heptagon" from our notation based on the proof, rather than an empirical fact.
• Lecture V: Wittgenstein delves deeper into the meaning of mathematical possibility, using the construction of the pentagon versus the heptagon as examples. He questions what Euclidean geometry states when it speaks of equal lengths without defining a method of measurement, suggesting it provides a groundwork or model for description rather than describing objects in the ordinary sense. The lecture proposes that a mathematical proof (like demonstrating how puzzle pieces fit or constructing a pentagon) doesn't show an empirical possibility but rather provides a paradigm or model, establishing a new criterion for what counts as "possible" within that mathematical system.  
• Lecture VI: The lecture examines the nature of mathematical axioms and proofs, questioning the meaning of "can" in axioms like "a straight line can be drawn between any two points". Wittgenstein explores the concept of following a rule "analogously" or "in the same way", arguing that our judgment of what constitutes the "same" way is based on shared training and practice, not an inherent property of the steps themselves. He concludes that mathematical proofs often establish or clarify the use of terms like "analogous", essentially teaching a technique or classifying things rather than discovering pre-existing relationships, comparing mathematical "discovery" to invention.  
• Lecture VII: Wittgenstein investigates the nature of proof and truth in mathematics, questioning the idea that a proof simply constructs a true proposition by corresponding with reality. He uses the example of correlating strokes in a hand figure with points on a pentagram to argue that mathematical proof establishes an internal relation between concepts by setting up a new criterion or paradigm, making the proposition "timeless" and independent of specific experiments. The lecture suggests that accepting a proof means accepting a new way of establishing equality or compatibility, changing the meaning of the terms involved.  
• Lecture VIII: This lecture contrasts finding a physical object (like a white lion) with finding a mathematical construction (like the heptacaidecagon), arguing the latter involves finding a shape satisfying internal conditions defined by the mathematical system, not discovering an external object. Wittgenstein refutes the idea of discovering mathematical facts like 125÷5=25, stating the result is part of the technique or calculus itself, which is invented, not discovered; its usefulness stems from extra-mathematical considerations. He further explores the meaning of symbols like negation ('~'), suggesting their meaning arises from their use within a system, not from inherent properties or fixed mental pictures.  
• Lecture IX: Wittgenstein examines proofs of impossibility, like constructing a 100-gon using only bisection from a square, suggesting such proofs function primarily to dissuade attempts by changing our idea of what the construction involves. He argues that mathematical problems like trisecting an angle arise from analogies in language ("bisect", "trisect"), and the proof of impossibility clarifies the limits of a specific technique, effectively excluding certain phrases (like "trisection with ruler and compass") from the notation. The lecture touches on the nature of mathematical belief and hypothesis, questioning how one can "believe" a theorem before it's proven, setting up the next discussion.
• Lecture X: This lecture scrutinizes the idea that mathematical calculation is a form of experiment, questioning what the "result" of such an experiment would be and how right/wrong applies. Wittgenstein argues that if a calculation is treated as an experiment to see what result rules lead to, it only works until the rules determine the result, after which it becomes a test of whether rules are followed. He concludes that calculation becomes a standard or norm (a "picture" deposited in archives), independent of any single experiment, based on widespread agreement in practice, not on discovering an objective result.  

Lectures XI-XX

• Lecture XI: Wittgenstein continues discussing calculation versus experiment, emphasizing that mathematical rules (like 2×2=4) are adopted and become standards, independent of empirical outcomes, unlike scientific hypotheses. He addresses the issue of infinite possibilities (like infinite multiplications), arguing that rules and paradigms, not an exhaustive list of instances, are what's archived or learned, and the concept of infinity pertains to the technique learned, not a huge quantity. The lecture concludes by reiterating that mathematical propositions function like definitions or grammatical rules, preparing symbols for use rather than describing an external reality.  
• Lecture XII: (The provided text skips from Lecture XI page 113 to Lecture XIII page 123. There is no content available for Lecture XII in the provided text.)
• Lecture XIII: Wittgenstein examines the nature of mathematical belief and proof, particularly regarding the recurrence of digits in division (like 1÷7). He questions what it means to "believe" a mathematical result before proof, suggesting it might mean believing a certain continuation is the most "natural" or that a particular rule will be adopted, rather than believing an objective fact. The lecture explores the idea of mathematical proof as shortening a process, arguing that accepting a shortcut (like using period recurrence instead of continuing division) involves adopting a new technique or criterion for correctness, influenced by consistency and practicality rather than discovering a pre-existing necessity.
• Lecture XIV: This lecture critiques the distinction between mathematical proofs that merely "convince" versus those that "really prove" something indubitable, arguing this stems from a confusion about the role of proof. Wittgenstein suggests that mathematical proof connects a proposition to a system, persuading us to adopt it as a rule within that system, rather than revealing an external truth for which the proof is mere evidence. He analyzes the idea of "mathematical reality" and "mathematical possibility," comparing it to chemical possibility (like H₂O₄), concluding that possibility in these contexts refers to sense within a given symbolic system or language, not a shadowy existence.
• Lecture XV: Wittgenstein continues exploring the idea of mathematical reality, contrasting finding a unicorn with finding the construction of a heptacaidecagon, emphasizing the latter involves creating a new analogy or projection within a system. He challenges the notion that mathematics describes a pre-existing realm by analyzing the "arbitrariness" of chess rules versus the non-arbitrariness of chess theory, suggesting the difference lies in the obvious application of mathematical theory, which chess lacks. The lecture discusses counting roots of an equation, arguing that how we count (e.g., saying a quadratic has two roots) is a convention or rule adopted for practical reasons within the mathematical system, not a discovery about the equation itself.
• Lecture XVI: Wittgenstein further examines counting in mathematics (like counting equation roots) versus ordinary counting (like counting people), stressing that the mathematical version involves establishing rules and conventions within a system. He critiques Russell's definition of number ("a class of classes similar to a given class"), arguing it merely substitutes one set of symbols/concepts for another without resolving the fundamental issue of how "correlation" or "similarity" is established or applied in any given case. The lecture concludes that Russell's logic doesn't provide a foundation for arithmetic because determining sameness of number (e.g., in large sets of variables) relies on pre-existing arithmetical techniques like counting, rather than Russell's logic providing the basis for them.  
• Lecture XVII: This lecture scrutinizes Russell's specific method for showing one-to-one correlation between classes of the same number using identity (e.g., the relation x=a∧y=b). Wittgenstein argues this "relation" is trivial and doesn't provide a practical method for establishing numerical equality between sets of actual objects (like apples in boxes), as it presupposes knowledge of identity and how to apply names, which itself depends on counting. He reiterates that while Frege's definition of number clarified the grammar (number as a property of a concept), it created new confusions regarding predicates and individuals, and ultimately doesn't determine how number words are actually used outside mathematics.
• Lecture XVIII: Wittgenstein discusses the idea that logical laws (like the law of contradiction) are "self-evident" or recognized by intellectual "inspection," arguing this is misleading because the statement's value depends entirely on its application and the consequences drawn from it. He explores what it means to assume the law of contradiction is false, suggesting it amounts to not having rules for how to react to contradictory orders or statements within our established language technique. The lecture proposes that explanations for why contradictions "don't work" (like T-F notation or mechanism analogies) are just alternative symbolisms or pictures, and the real reason we exclude contradictions is the practicality and established conventions of our language use.
• Lecture XIX: This lecture elaborates on the relationship between the meaning of a word (like "not" or "all") and its use, arguing that while mental pictures or initial explanations often guide usage, the meaning is ultimately defined by the ongoing, shared practice or technique. Wittgenstein contends that logical laws like '(x).fx entails fa' are not based on discovering inherent meanings but represent the natural, conventional continuation of the techniques we learn for using words like "all"; violating these laws means using the words differently. He critiques the idea of a "logical machinery" behind symbols, arguing it's a misleading metaphor based on using physical mechanisms as symbols for behavior, whereas in logic, the rules and our adherence to them are the "mechanism".
• Lecture XX: Wittgenstein challenges the idea of a "super-rigidity" in logic, comparing it to the misleading notion of a kinematic rod being perfectly rigid (whereas it simply has no property corresponding to expansion/contraction in the calculus) or a law being "super-inexorable" (arising from linguistic parallels, not experience). He argues that the perceived inexorability of logic stems from confusing the rules of the calculus (which are fixed by us) with descriptions of reality; logic's "hardness" comes from our decision to adhere to its rules as a standard. The lecture concludes by stating that denying the existence of a "logical mechanism" or "super-rigidity" means showing these ideas arise from misleading pictures and analogies, not from the same source as ordinary rigidity.

Lectures XXI-XXXI

• Lecture XXI: Wittgenstein questions how we become convinced of logical laws, rejecting the idea that they are corroborated by experience (like the law of contradiction or 2+2=4). He suggests recognizing logical laws amounts to adopting and following certain linguistic practices and techniques because they align with our natural inclinations and avoid pointlessness or confusion, comparing this to the rejection of "reddish-green". He analyzes the Liar Paradox ("I am lying"), arguing its puzzling nature stems from treating it like a meaningful proposition within a useful system, whereas it's simply a useless language-game arising from grammatical analogy, and the resulting contradiction doesn't invalidate logic itself.
• Lecture XXII: The lecture directly addresses Turing's concern that contradictions in a logical system used for applications (like building bridges) could cause failures. Wittgenstein argues that bridge failures stem from incorrect physics (wrong natural laws) or mistakes in calculation (applying the calculus wrongly), not from contradictions inherent in the mathematical/logical calculus itself. He contends that discovering a contradiction means we might need to refine the rules or avoid using that specific part of the calculus, but it doesn't necessarily invalidate prior correct applications or mean the system is inherently flawed or dangerous as long as the contradiction isn't actively used.  
• Lecture XXIII: Wittgenstein continues discussing the perceived danger of contradictions, suggesting that a "hidden" contradiction only becomes problematic if it leads to unintended pathways or ambiguities in applying the calculus, comparing it to an unnoticed escape route in a prison. He argues that finding a contradiction like Russell's paradox doesn't necessarily vitiate the entire system (like Frege's logic) for all its uses, especially if we simply avoid the specific problematic formation (like ϕ(ϕ)) or know how not to proceed from the contradiction. The lecture emphasizes the importance of understanding how we might get into trouble, suggesting it's often about misapplication or unintended rule interpretations rather than the mere presence of a potential contradiction.
• Lecture XXIV: This lecture focuses on logical laws as "laws of thought," proposing they represent our established practices and techniques for using language and making transformations, rather than describing external facts or psychological processes. Wittgenstein compares logical laws like the law of contradiction to synthetic a priori statements like "a patch cannot be both red and green," arguing both reflect our inclination to continue using concepts in certain "natural" ways and to exclude combinations that upset our system or seem pointless. He critiques the idea of two types of proof (one merely convincing, one truly grounding), suggesting mathematical proof persuades by making connections within a system, and accepting a proof means adopting that technique or rule, not confirming an external truth.
• Lecture XXV: Wittgenstein further analyzes the idea that mathematical propositions correspond to a "reality," arguing this comparison to physics is misleading. He distinguishes between a reality corresponding to a true sentence (affirming the sentence) and a reality corresponding to words (explaining their meaning/use via grammar), suggesting mathematical propositions are more like the latter – they function as rules or grammatical preparations for language use. He emphasizes that mathematical propositions are not "about" numbers in the same way experiential propositions are about objects; rather, they develop the calculus that gives number words their meaning for application outside mathematics.
• Lecture XXVI: Wittgenstein continues exploring the relationship between mathematics, logic, and reality, reiterating that mathematical propositions primarily establish rules and prepare symbols like number words for their application, rather than describing mathematical entities. He critiques misleading imagery associated with mathematical concepts like infinity or higher dimensions, arguing that their meaning comes from their specific, often pedestrian, use within a calculus, not from analogies suggesting vastness or mystery. The lecture cautions against interpreting the "depth" or "beauty" of mathematics as residing in some profound meaning beyond the calculations themselves, attributing such feelings to potentially misleading pictures and analogies.
• Lecture XXVII: Wittgenstein examines the relationship between logic and arithmetic, arguing that arithmetic does not rest on logic (like Russell's) because determining the validity of logical transformations (like tautologies involving large numbers of variables) itself presupposes arithmetic abilities like counting and comparing numbers. He suggests that Russell's logic is just one possible calculus or method, comparable to other ways of counting or calculating, and we trust our established arithmetic practices over it in case of conflict. The lecture highlights that Russell's definitions connect arithmetic concepts to logical ones (like addition to disjunction), which clarifies some aspects but doesn't provide a unique foundation or dictate the specific rules of calculation we must adopt.
• Lecture XXVIII: Wittgenstein reiterates that logic isn't a foundation for arithmetic in the sense that arithmetic uniquely follows from it; different arithmetics could be developed alongside the same logic. He critiques the Russell/Frege use of logical notation like (∃x).φx or (x).φx as potentially confusing because it treats concepts like "man" or "circle" as simple predicates of bare individuals ("x"), obscuring the varied grammar and criteria involved in applying these terms in ordinary language. The lecture emphasizes that mathematical methods evolve like methods of measurement, introducing new techniques and meanings (e.g., for large numbers or complex logical formulae) rather than relying on a single, fixed logical base.  
• Lecture XXIX: This lecture explores what logical propositions (tautologies like p . p⊃q .⊃ .q) "say," concluding they "say nothing" in the sense of not providing empirical information; their point lies in demonstrating rules of inference or the structure of our language use. Wittgenstein argues that asserting a tautology is akin to stating a rule for transforming sentences, showing how certain combinations cancel out information. He suggests logic could even be done by proving non-tautologies (showing certain inferences cannot be made), highlighting that the form of logical propositions is less important than their use in structuring reasoning and calculation.
• Lecture XXX: Wittgenstein discusses the equivalence (n)φ . (m)ψ . Ind . ⊃ . (n+m)φ∨ψ as a potential logical definition of addition, arguing it fails because it doesn't inherently yield arithmetic. He points out that verifying this as a tautology for large numbers requires a pre-existing method of calculation (like arithmetic addition) to compare the number of terms, meaning arithmetic isn't based on this logical form but vice-versa. The lecture stresses that we trust our established arithmetic calculations over alternative methods like Russell's logic or direct correlation, using calculation as the standard to check other methods, not the other way around.
• Lecture XXXI: Wittgenstein examines proof by mathematical induction, questioning the certainty derived from proving a base case @(1) and the inductive step (n):@(n).⊃.@(n+1) to conclude (n).@(n) without performing every step. He argues this "shortcut" is not about magically covering infinite steps but about adopting the inductive proof itself as a new rule or criterion for the result (e.g., @(3000) = ψ(3000)). This rule is justified by empirical facts like general agreement in calculations, and it becomes the standard against which deviations (like getting a different result from 3000 steps) are judged as mistakes.  

Quotes

Lecture 1

What kind of misunderstandings am I talking about? They arise from a tendency to assimilate to each other expressions which have very different functions in the language. We use the word "number" in all sorts of different cases, guided by a certain analogy. We try to talk of very different things by means of the same schema. This is partly a matter of economy; and, like primitive peoples, we are much more inclined to say, "All these things, though looking different, are really the same" than we are to say, "All these things, though looking the same, are really different." Hence, I will have to stress the differences between things, where ordinarily the similarities are stressed, though this, too, can lead to misunderstandings.

- p. 15

Suppose Smith tells the municipal authorities, "I have provided all Cambridge with telephones - but some are invisible." He uses the phrase "Turing has an invisible telephone" instead of "Turing has no phone."

There is a difference of degree. In each case he has done something but not the whole. As he does less and less, in the end what he has done is to change his phraseology and nothing else at all.

To think this difference is irrelevant because it is a difference of degree is stupid.

Lecture 2

Should you... say, "I believe that I intend to play chess, but I don't know. Let's just see" - ? Just as Russell once suggested that we don't know what we wish, don't know whether we want an apple or not.

Suppose we said, "What he said was just a description of his state of mind." But why should we call the state of mind he is in at present "intending to play chess"? For playing chess is an activity...

One might say, "Intending to play chess is a state of mind which experience has shown generally to precede playing chess." But this will not do at all. Do you have a peculiar feeling and say, "This is the queer feeling I have before playing chess. I wonder whether I'm going to play"? - this queer feeling which precedes playing chess one would never call "intending to play chess."

...I have been considering the word "intend" because it throws light on the words "understand" and "mean". The grammar of the three words is very similar; for in all three cases, the words seem to apply both to what happens at one moment and to what happens in the future.

Suppose I teach Lewy to square numbers by giving him a rule and working out examples. And suppose these examples are taken from the series of numbers from 1 to 1,000,000. We are then tempted to say, "We can never really know that he will not differ from us when squaring numbers over, say, 1,000,000,000. And that shows that you never know for sure that another person understands."

But the real difficulty is, how do you know that you yourself understand a symbol? Can you really know that you know how to square numbers? Can you prophesy how you'll square tomorrow? - I know about myself just what I know about him, namely, that I have certain rules, that I have worked certain examples, that I have certain mental images, etc etc. But if so, can I ever know if I have understood? Can I ever really know what I mean by the square of a number? Because I don't know what I'll do tomorrow.

- p. 27

Does the formula $y=x^{2}$ determine what is to happen at the 100th step?

It might mean, "Is there any rule about it?"

If it means, "Do most people after being taught to square numbers up to 100, do so-and-so when they get to 100?", it is a completely different question. The former is about the operations of mathematics but the latter is about people's behavior.

-p. 29

We have all been taught a technique of counting in Arabic numerals. We have al of us learned to count - we have learned to construct one numeral after another. Now how many numerals have you learned to write down?

Turing: Well, if I were not here, I should say ${\mathfrak {N}}_{0}$

...I did not ask, "How many numerals are there" This is immensely important. I asked a question about a human being, namely, "How many numerals did you learn to write down?"

Lecture 3

This is a part of our ordinary speech; it is no more a mathematical proposition than "You have on a number of shoes which satisfies the equation $x+5=7$" is a mathematical proposition.

- p. 34

Watson: ...for instance, suppose one said, "There are in this room as many people as the number of moves which are needed for Black to checkmate White from such and such a position," would that be called an application of chess?

Wittgenstein: Why, certainly it would be. Indeed it might be the case that we discovered a fixed correlation between the number of moves needed to checkmate from certain positions and the number of, say, atoms in certain molecules. Then, in order to discover the number of atoms in such and such a molecule, it might be easiest to set the chessmen in such-and-such a position and play chess. That would certainly be an application of chess.

- p. 34

We call these things proofs because of certain applications; and if we couldn't use them for predicting, couldn't apply them, etc, we wouldn't call them proofs. The word "proof" is taken from ordinary everyday language, and it is only used because the thing proves something in the ordinary sense.

- p. 38

There is no "general proof." The word "proof" changes its meaning, just as the word "chess" changes its meaning. By the word "chess" one can mean the game which is defined by the present rules of chess or the game as it has been played for centuries past with varying rules.

We fix whether there is to be only one proof of a certain proposition, or two proofs, or many proofs. For everything depends on what we call a proof.

It is not the case that there are two facts - the physical fact that if one counts the squares one gets 756 and the mathematical fact that 21 times 36 equals 756.

- p. 39

Lecture 4

The point is that the proposition 25 x 25 equals 625 may be true in two senses. If I calculate a weight with it, I can use it in two different ways.

First, when used as a prediction of what something will weigh - in this case it may be true or false, and is an experiential proposition. I will call it wrong if the object in question is not found to weigh 625 grams when put in the balance.

In another sense, the proposition is correct if calculation shows this - if it can be proved - if multiplication of 25 by 25 gives 625 according to certain rules.

...It is of course in the second way that we ordinarily use the statement that 25 x 25 equals 625. We make its correctness or incorrectness independent of experience. In one sense it is independent of experience, in one sense not.

Independent of experience because nothing which happens will ever make us call it false or give it up.

Dependent on experience because you wouldn't use this calculation if things were different. The proof of it is only called a proof because it gives results which are useful in experience.

We often put rules in the form of definitions. But the important question is always how these expressions are used.

Suppose someone knew logic but not mathematics. Could we teach him to multiply simply by definitions? Can the decimal system be taught by definitions? If Russell can do all mathematics in Principia Mathematica, he ought to be able to work out 25 squared equals 625. But can he? How could decimal numbers be introduced into Principia Mathematica? Russel and Frege said that by introducing some more definitions into their systems they could prove such things as 25 squared equals 625. But we cannot teach anybody to multiply by definitions.

Mathematics and logic are two different techniques. The definitions are not mere abbreviations; they are transitions from one technique to another.

...It is immensely important to realize that definitions join two quite different techniques. Sometimes the difference is important and sometimes it is trivial (as when we write c instead of a times b). But the fact that the difference is trivial should not blind us to the fact that these are two different techniques.