The posthumously edited and compiled book Philosophical Grammar, by Ludwig Wittgenstein, is divided into two parts, one on "the proposition and its sense" and one "on logic and mathematics", with an appendix between the two. The book is a "snapshot" of his system as it was at one time, about 1932, in its development. Those who have studied Wittgenstein will be familiar with the interplay between these two topics in his thought.
In this short essay, I have concentrated on the foundations of mathematics and the concept of proof, which will also be recognized by those who have studied Wittgenstein as topics about which he wrote much. We will find that proof is the same thing as a historical record or a set of instructions. Mathematics must be seen as a whole in order to understand the meaning of the search for its foundations. As for the form of this essay, it consists of a series of quotes from the book, each followed by a short commentary or meditation.
Beyond the scope of this essay, but worthy of note, are passages in the book dealing with the distinction between different senses of "try" and "attempt", with the observation that algebraic proofs cannot be gotten out of arithmetic proofs, and with "understanding" and "describing the application of" a word and the way our form of life meshes with it. This lattermost topic would be of keen interest to those who have done research on the notion of "following a rule".
Ann Arbor, Michigan
October, 1990
A foundation that stands on nothing is a bad foundation.
(pg. 297)
Wittgenstein is here referring to the idea that logic and math are based on axioms. By saying that they are a bad foundation, he wants to say that they are no foundation at all. This passage by itself cannot be taken to mean that Wittgenstein believes in no foundation what so ever. We must examine further passages for an answer to this question.
...what we have to do is to describe the calculus...give its rules and by doing so we
lay the foundations of arithmetic.
(pg. 297)
The foundation of math is the description of math. A vision of the whole is the basis for the step-by-step construction of the whole. Wittgenstein speaks of having an "overview" or a "bird's eye view" of mathematics as a whole. Because we are concerned with justifying the whole of mathematics, with finding its foundations, we must continually look at the whole. It is not easy to see this whole, and the question may even be raised whether or not all of mathematics is an organic whole. But if we can see all of math as a whole, then we will have come closer to understanding what its foundations could be. We may have to include or exclude unexpected things to see math as a whole; in this quote, Wittgenstein speaks of arithmetic, but we can generalize to math in general.
Teach it to us, and then you have laid its foundations.
(pg. 297)
The foundation is the possession of the ability and knowledge required to work within mathematics. To know how to calculate is to have the foundations of a calculus - Wittgenstein speaks here of mathematics as a whole as a calculus - recall that many textbooks are called "foundations in spelling" and "foundations in math" and "foundations in reading". And here we note that a proof in Euclidean geometry is that same as instruction in Euclidean geometry. If I ask, "show me a proof that you have bisected an angle," I will get exactly the same answer as if I had asked, "show me how to bisect an angle."
I want to say the place of a word in grammar is its meaning.
(pg. 59)
That which is for a mathematical calculus its foundation is for a grammar the meaning of its words. The meaning of a word is its relation to other words, its position in the structure of language - always mindful that here we might be speaking of "deep" and semantical structure and grammar, and not necessarily of syntactical and "surface" structure and grammar of natural language. Or are we? The meaning of "apple" is expressed in part in that it can be used with words like "green, yellow, read, sweet, big, small" but not with words like "jazz, waltz, B flat, F sharp" - so we see that the meaning of the word is its place in the grammar; also, parts of speech, noun, pronoun, verb, adjective, etc., show us a word's place in grammar. But when we say that "apple" is never preceded by "a" but is sometimes preceded by "an", is that a grammatical proposition which also shows us meaning? (And in math, "1 divided by 0" or "I am pointing to the point on the number line which is i, the square root of -1", which are both grammatically incorrect, yet we want to say that we know what the speaker meant.) Some rules of grammar seem completely isolated from meaning. Is there a pure semantics or a pure syntax? In a natural language, some linguists argue that to say "an apple" or "a apple" carries meaning about the education of the speaker and so forth. Just as a word draws its meaning from grammar, a formula or equation draws its meaning, truth, significance, and foundation from the calculus or from math as a whole. A rule about the use of a word, the word's place in grammar, is a mixture of syntax and semantics; the two are difficult or impossible to separate. And statements in math which violate the grammar (syntax) of math also contain semantic errors, even if we have a psychological idea about what might have been meant. The unintended meaning of someone who says "a apple" (instead of "an apple") is not relevant to us here at all, because it does not deal with language as such, but rather an attempt to know the mental state by observation of behavior - assuming that the error was unintended, we learn that the person does not know the rule, just as we would know that a person does not know how an automobile works, if we saw him putting water instead of gasoline into its tank. If a person chooses to say "a apple" then that is simply the same as saying "I choose not to follow the rule at this time". It may come to pass that we find the distinction between syntax and semantics to be artificial and non-existent. Or it may be that the distinction is real, but that the two are inseparable. Or one may be (temporally or logically) prior and the other posterior, or stand in a causal relation. But if we may state one thing clearly and affirmatively, it is that these two are not in isolation from each other, that there is some relationship between them.
The meaning is the role of the word in the calculus.
(pg. 63)
This calls to mind Quine's "rabbit / gavagai" example. If language is a calculus, what is the result? Is Wittgenstein here influenced by behaviorism? But there is more than mere behaviorism - let us compare the role of a mathematical term in a calculus. What is the role of pi in calculating the circumference or area of a circle? What is the role of the division symbol in a problem like "four divided by three equals one and one-third"?
...we have a grammatical structure which cannot be given a logical foundation. (pg. 304)
Wittgenstein seems to want to say that we can give a foundation to our grammatical structures (natural languages, mathematics, chess), but that this foundation cannot be a logical one. And this points us to a question which is so large that very few people ask it; we are always looking at the trees and have not seen the forest. What is a foundation? Why do we want a foundation for mathematics? What will we gain from this foundation? For philosophers in recent times, this search for a foundation of mathematics is like a quest for the Holy Grail, which has been continuing for several generations, and we have joined it in progress, without having witnessed the spectacles which launched this search, now more than a century ago. These new-comers are people like myself who entered the world of contemporary philosophy in the early 1980's. This search for a foundation began with fears about the certainty of mathematics. But what was the threat? What was the danger? This search for foundations began before Godel attacked Russell's system, and before Russell attacked Frege's work. As far back as Hilbert, the search for foundations was already in progress. We might even say that this went as far back as Euclid. He was not content with merely doing geometry and making a catalogue of manoeuvres such as "bisecting an angle" and "bisecting a segment" and "creating a line parallel to a given line." Beyond that, he wanted to show that they were derived in a truth-preserving manner form obviously true axioms. Euclid wanted to show that all of these moves are always true and can be used in any conceivable combination or situation. He wanted to have a foundation so that if someone doubted that he had really bisected an angle, he could prove that he had. But the proof is really no more than the instruction how to do a certain geometrical move. When I say "prove it," I am saying, "show me how you did it." When I ask for a proof, I want to know that you did it right. And a proof requires foundation. Every demonstration requires a foundation. Math needs a foundation so that you can use the foundation as the basis of your demonstration, so that you can prove to me that you calculated correctly. The search for foundations became more appealing when mathematics ventured into topics in which there was little sense of "intuitively" correct answers: answers could only be seen correct in a technical sense, not an intuitive sense (cf., e.g., various geometries).
Whether [a proposition] is provable or not depends on whether we treat it as
calculable or not. For if the proposition is a rule, a paradigm, which every calculation has
to follow, then it makes no more sense to talk of working out the equation, than to talk of
working out a definition.
(pg. 395)
Wittgenstein here wants to say that we cannot prove the rules of our game. The axioms, formation rules, etc., are never proved within the game. A move like bisection is proved and provable. We do not prove A=A, but we use it. We use "A=A" to prove "4=4". We prove
x = a(b + c)from
x = ab + ac
using moves, but we do not prove the moves. Nor do we prove that
x = ab + ac
Wittgenstein speaks also of definitions. Using definitions, I can prove that "a bachelor has
no wife," but I cannot prove that "a bachelor is an unmarried male." Within a calculus,
we can prove. In particular, a inductive proofs are only a "check" of a rule, and thus a
proof in a limited sense.
Whether a pupil knows a rule for ensuring a solution to sin x.dx is of no interest; what
does interest us is whether the calculus we have before us (and that he happens to be
using) contains such a rule.
(pg. 379)
What is important is if and how an answer can be given. Remember that a proof is a demonstration of how something was done. If I ask a man to fix my automobile, I am interested in his knowing how to fix my car - I am a mathematician. If I am buying a car, and its workings are so complex that I cannot understand them (or don't have the time to), but I know that it is possible that it will need repair in the future, I am interested in whether it is in principle possible for the automobile to be repaired - I am a philosopher, I am a meta-mathematician. The manufacturer shows me that any situation that may come to pass with the car is repairable - the foundations of mathematics. But how does he show me? How far can we go with the analogy of the car? This statement is attributed to Wittgenstein, but I have not researched the textual evidence: "there is no metamathematics."
The philosopher easily gets into the position of a heavy-handed manager, who,
instead of doing his own work and merely supervising his employees to see they do their
work well, takes over their jobs until one day he finds himself overburdened with other
people's work while his employees watch and criticize him. He is particularly inclined to
saddle himself with the work of the mathematician.
(pg. 369)
Because a philosopher needs the results of other specialists, he is often tempted to do work in other fields: math, history, physics, psychology, literary criticism, and art. The philosopher needs to know that in principle a proof or calculation is possible, but how it is done does not interest him. Again, he does not need the results of a calculation, although he might need to know what their logical form or category is. A man who brings his car in for repair often says, "I don't care what or how you do it, just fix it!" or at most, "Do what you want, just fix it and keep the repairs inexpensive!" As any business man knows, the art of delegating responsibilities to the appropriate specialists is vital. One need only know that they can do their work well. Calculation is not the job of the philosopher. He needs only to know in principle. The philosopher needs only to know that "in geometry we have some finite number of axioms, and given these axioms, some things are possible, others impossible, and others necessary." To know or learn what is possible, or what kinds of calculations there are, or to know and learn the axioms and their number, is a waste of time. This is true for history and all other disciplines. To know the wars and kings is not important. We need only know that there were and are wars and kings. Thus we deal metaphilosophically with any specific body of knowledge.
Now it is part of the nature of what we call propositions that they must be capable of
being negated. And the negation of what is proved also must be connected with the proof;
we must, that is, be able to show in what different contrasting conditions it would have
been the result
(pg. 376)
This is closely related to the reductio ad absurdam. Since the form of the proof is dictated by the form of what is proved, and because whatever is proved could be negated, then the proof can be altered to show what this negation means. And because the altered proof shows what the negated proposition means, then we may conclude that the proof shows what the proposition means. And here we are close to saying that the proof is the meaning. Although I have not researched the textual evidence, this statement is attributed to Wittgenstein: "If you want to know what is proved, look at the proof."
Tell me how you seek, and I will tell you what you are seeking.
(pg. 370)
A proof is a seeking, a search. A proof is further a successful search. If I seek to bisect an angle, and I write down each step I take, and if at the end of my search I am successful, then that written record is a proof. A proof is showing how someone attempted to do something. If I watch a man in a workshop, I will be able to know what he is trying to do. Thus if someone were to ask, "what is the bisection of an angle?" I would answer him by showing him the proof. A proof is nothing more than a historical documentation of a search, or a set of instructions, until I call it a proof. The ledgers and account books of a business are only historical records, and will remain only that, unless or until the business is audited; then they become "proof". And the booklets put out by the government which tell us how to pay taxes are instructions and will remain only that, unless or until they are used in court, and then they become "proof". So also with Euclid: when he wrote how to bisect an angle, it was first merely a record, so that he would know and remember how he did it, and then later an instruction, to show others how to do it. The category "proof" is thus subjective, one which we project onto these things. Thus "proof" is an unstable "foundation".
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