This page is written from the viewpoint of Wittgenstein's logic of language, but it may not represent Wittgenstein's own views about the philosophy of mathematics, in many respects.
Philosophy of Mathematics
Outline of this page ...
Grammatical myths | Mathematics versus Philosophy
At school we were taught that mathematics is about the properties of numbers and that numbers are abstract objects. How we were to verify this (how to read off from the object its properties), we were not taught. Perhaps we thought that if only we knew more mathematics.... What we were taught at school were grammatical myths -- fanciful inventions about the meanings of mathematical signs that have no relation whatever to how those signs are used. And knowing more mathematics will not enlighten us here, because this is a philosophical not a mathematical problem.
The sign '2/x' is an instruction telling us to divide 2 by whatever number 'x' is replaced with. But replace 'x' with '0' and we do not know how to go on. We ask our teacher who answers that "division by zero is impossible". But what kind of impossibility is this? The impossibility is grammatical -- i.e. the sign '2/0' is an undefined combination of signs (in just the way that '2/+' is). In a word, division by zero is only impossible because 'division by zero' is undefined (language); that is the only meaning 'impossible' has here. [Note 1]
But how do I know? Because I only talk about what I know -- about the use of mathematical signs that I learned at school. And what I do not know does not concern me here, because: what a sign is used to do in a particular context is the sign's meaning in that context. (Wittgenstein's Lectures on the Foundations of Mathematics p. 13-4). What I can describe are some ways that mathematical signs are used, namely, the ways I and many other people were taught in our early school years. I won't try to talk about anything beyond addition, subtraction, multiplication, division and elementary algebra -- because that is the mathematical language I still remember and these are still the techniques that I use in my daily life.
The Philosophy of Mathematics is not concerned with this or that particular mathematical calculus (metaphorically: "game governed by strict rules"), but with the nature of mathematical calculi as such. But to investigate these calculi, it must look at particular calculi, and it may discover that all the things we call 'mathematical calculi' do not have a common nature. But I know nothing about that; I myself am acquainted with very few calculi and not at all with what is called "higher mathematics". All that concerns me here is trying to understand the nature of the mathematical language I learned at school -- e.g. the ways it is like or unlike natural language. School left me and many other people with a lot of confused ideas about the nature of mathematics, and one task of philosophy is to clear up that confusion.
The false account of the grammar (meaning) of mathematics
When we are taught to use numbers at school we are taught ostensively. E.g. we are trained to use the number-sign '2' (i.e. the number 2) by being taught to add and subtract chalk lines. But if we want to know about the bearer of a name, we have to ask -- not for a definition of a word, but to see the name's bearer. And chalk is not as it were the bearer of the name '2'.
And this has been pointed out by countless philosophers -- but not in order to justify the evident conclusion, i.e. that numbers are not names of objects, -- but in order to justify the conclusion that numbers are invisible objects! As always: "If we are making sense, then we must be talking about some object, if not a visible one then an invisible one ..."
The grammatical account offered here, of course, is that numbers are not names. That what we can know about e.g. two and zero is completely given to us in rules for using the signs 'two' and 'zero'. That there is no bearer of a name to ask about here.
*
"Conceptual possibility"
Consider that while perhaps no one will say that division by zero is empirically impossible, some philosopher will undoubtedly say that it is conceptually impossible. 'If you know the meaning of division, then you know that a number cannot be divided by zero; for division is division into parts, but division by zero would be division into no parts -- which is not a division at all.' But then why isn't division by one impossible -- for "division into one part" is not a "division" either? Why isn't '2/1' a meaningless combination of signs? Only because there is a rule that tells us what to do with it. (And if there were no rule, one could be invented.)
'x/1 = x' is a rule for using mathematical signs. It is not as it were an invitation to contemplate "the conceptual possibility of division by oneness".
In logic the word 'concept' simply means: a sign with a use in our language -- i.e. a grammar. But above we encounter the picture of a concept as an "abstraction" that one somehow grasps the meaning ("abstracts the essence", without bothering to ask whether or not there is one) or nature of: "If you know the meaning of division ..." You go and look at "concrete objects". But you don't go and look at "abstract objects"; you just sort of -- think about them.
That in a nut shell (which is very often taken for a palace with seven towers) is the philosophical tradition that Wittgenstein set out to overturn; -- it is an obscurantism, a path into complete darkness. It is not an "alternative logic" -- it is no logic at all, because it entirely negates objectivity by placing meaning in the realm of "whatever seems right" (Philosophical Investigations § 258).
I am in a sense making propaganda for one [way] of thinking as opposed to another. I am honestly disgusted with the other. (LC p. 28)
Non-natural properties and Russell's view of language
Moore wrote that "Goodness is a non-natural property". And most philosophers write as though they thought meaning to be a non-natural property of language, as though it were something that only the clever -- Bertrand Russell's "educated" -- could perceive. This way of thinking leads only to self-mystification and to the mystification of one's students.
Even if it were nothing else, this would be a motto worthy of a philosopher: "Whatever can be put into words can be put clearly". But how can we clarify the meaning of statements said to result from inscrutable processes (-- what else are perceptions of non-natural properties or abstract objects? --) said to go on in "the inner night of the soul" (Hegel's expression) -- for their meaning is then out of public reach and hence public criticism?
W.E. Johnson, the logician, said to Drury (ca. 1929) ...
I consider it a disaster for Cambridge that Wittgenstein has returned. A man who is quite incapable of carrying on a discussion. If I say that a sentence has meaning for me, no one has the right to say it is senseless. (Recollections p. 103)
What right had Wittgenstein? None -- if sense and nonsense is subjective. But in Wittgenstein's logic of language it is objective. That is the grammar Wittgenstein chose for 'sense' and 'nonsense'. In his logic of language, a "meaning" that cannot be stated in rules for using a word is no meaning at all; and common natures, abstractions, or, essences, must be demonstrated, not assumed, to exist.
Nothing causes more confusion in philosophy than that one notion: "abstract object of thought".
*
Mental processes and the meaning of mathematical language
"A domicile is an important thing," Dostoyevsky's old prince says in A Raw Youth. This is that picture: "If a thing exists, mustn't it exist in some place? And therefore, if not in some visible place, then in some invisible place." And the related picture: "Words are names of objects, and the meaning of a word is the object the word stands for (i.e. the object the word names). Now then, how can a word stand for an object if the object itself does not have a home (i.e. a place to exist)?" [Note 2]
If it is not claimed that mathematical objects exist in an empyrean (a spirit world or the mind of God), it will be claimed that they exist in the human mind. "The meaning is the thought" or "the meaning is the thought process" appears to give the abstraction an, as it were, concrete dwelling place. But Wittgenstein's remarks apropos of how mathematicians work show that this is not what we mean by the language of mathematics (nor is it what we mean by the language of mind).
We are always being told that a mathematician works by instinct ... that he doesn't proceed mechanically like a chess player ... (Philosophical Grammar ii § 11, p. 295)
A chess player has the chessboard and all the chess pieces and, as it were, all the rules of the game directly in front of him, and if he announces 'Checkmate in seven moves' he must show us move by move how checkmate can be achieved in accordance with the rules of the game. If he "proceeds mechanically" -- by definition -- his demonstration to us will be a mirror image of the mental processes that lead to his announcement.
But the rules of chess are fixed once and for all, whereas we are told that mathematicians are always working to invent new mathematical grammars (rules for using mathematical signs). And reportedly their thought processes are not like those of the hypothetical chess player who thinks move-by-move. We are told that mathematicians "work by instinct" (or "intuition").
The "account books" of mathematics
But we aren't told what that is supposed to have to do with the nature of mathematics ...What I check are the account books of mathematicians; their mental processes ... as they go about their business ... are no concern of mine. (ibid.)
Neither are the mental processes -- whatever they may be -- of actual chess players of any concern to logic. What concerns logic is the demonstration that grammatically justifies the player's announcement. That is what Wittgenstein meant by his metaphor "the rules of the game": the rules are public. (The auditor pays no attention to what is said about the business; the auditor looks only at what is recorded in black and white, red.)
When we add, for example, 693 to 479, most of us have to do it step by step: 3 + 9 = 12, write 2 in the ones column, carry 1 over to the tens column, and so on. But there are human beings who have only to glance at 693 + 479 to tell us instantly that the sum is 1172. They do not appear to calculate; -- perhaps we shall want to say that they "work by instinct" (cf. PI § 236).
Mathematical signs -- like the signs of any other language -- have a grammar. And if we add 693 to 479, what justifies our result are the grammatical rules of arithmetic -- regardless of the mental processes by which we arrive at the result.
So that, if a mathematician invents new uses for mathematical signs or new combinations of mathematical signs, the meaning of those signs is not the mental processes by which they were invented or which may accompany their use, but the rules that now govern their use. Those rules are the "account books" of mathematics. Account books are objective.
The notion that the mental processes of mathematicians (or of anyone else) are the meaning of mathematical language is another grammatical myth. For mathematics would be completely unaffected if mathematicians had no mental processes at all.
When we don't know how to go on, then we need a rule
If A x C = B x C then A = B. But then if C = 0, any number can be proved to be equal to any other number. But we do not want our mathematics to be that way. So we make the rule 'But C may not equal zero' (WLFM p. 221-2), and further, that 'if A ≠ B, then neither A nor B may equal zero' -- where 'may not' means: is not permitted by the rules of the algebra "game".
But why? Is it because something is inherently wrong with a contradiction, e.g. 4 = 5? Or is it because we want our mathematics to have various applications outside itself -- as e.g. we want to be able to measure lengths of wood or amounts of money (we don't want $4 = $5).
But if we allowed the rule 'C = 0', would that be the end of mathematics -- or would it just be a different mathematics? 'C = 0' is a rule mathematicians resort to at the end of the work day. Why not? Because can't we imagine a people who only used maths equations as decoration or maths as we use chess -- i.e. as games with no subject matter outside themselves?
Confronted with a contradiction, or any other undefined sign, the student does not know how to go on. But what do we do when we are confronted with a contradiction? We proscribe it, and move on. We make a rule: 'but C may not equal zero', or 'but 2/0 is undefined'.
When we teach a child chess, we give it the rule: the bishop only moves along diagonals. Suppose we play a game and the child moves its bishop along a diagonal and clear off the board (the child says it wants to protect the bishop from attack). We now state the rule: but the bishop must remain on the board. But would it be the end of chess if we allowed the child's move?
The King's paper crown
Chess with Pilgrim's Option
Player may send his bishop on pilgrimage -- i.e. make it not subject to attack. But while he may move his bishop according to normal rules while the bishop is a pilgrim, he cannot use his bishop to attack while it is a pilgrim. He must use another turn to make the bishop a combatant again. (While the bishop is on pilgrimage, he must wear a paper hat; that is a meaning for The Blue Book p. 65: "I want to play chess, and a man gives the white king a paper crown ...")
Chess with King-at-Prayer Option
If his king is checked, player may declare the king to be at prayer (See Hamletiii, 3) and so not subject to attack, and then he may move another piece. But the king must rise from prayer for player's next move -- i.e. now the king may be in check and player must follow the normal rules. Like castling, a player may do this only once.
Maths as wall decoration (seeing as)
Wittgenstein suggested to imagine a people whose only use for mathematics is that they decorate the walls of their rooms with equations. (Zettel § 711; WLFM p. 34, 36, 39) Perhaps when this people reached a corner of the room, the rule that 'C may equal 0' would be useful to them.
Grammar and "Now I can go on" (PI § 151) in Maths
But it would be a mistake to conclude that mathematics consists entirely of rules. For example, '693 + 479 = 1172' is not a rule, but an application of rules. What we can say, however, is that whenever we are in doubt about how to go on in mathematics, what we need to resolve our difficulty is always a rule -- and only a rule, never a fact (about anything other than grammar). Because the subject matter of mathematics is not the facts about any objects. Mathematics, like chess, has no independent subject matter.
What are Numbers?
The grammar of the number-sign '2' -- or, the rules for using the number 2; the two forms of expression mean exactly the same, -- is no more that of name-of-object than the grammar of the sign 'elf' is that of name-of-object. So I have said, but how do I know?
There is no difference between asking whether numbers -- or the points, lines and planes of geometry -- might not after all really exist as spirits and asking the same thing about elves. If a notion or picture is unverifiable, it is because we ourselves have made it unverifiable (Z § 259); language answers to no one else. Which is to say that we have not chosen to connect the grammars of 'number' and 'verification'; so that the question-sign 'Is the number 2 really a spirit?' is not unanswerable but nonsense (undefined).
In this context the distinction between numbers and numerals creates confusion. Numerals, we are told, are mere signs -- but numbers are the subject matter of mathematics. We might not think that a simple distinction between the Roman and Arabic notations could do much harm. But imagine if we made a distinction between words and "word-als", e.g. 'book' and 'BOOK' would "stand for" the same word but be different "word-als". Then we might be inclined to imagine school-grammar to be the study of abstract objects called 'words'. So suggestible we are.
What are numbers? Words. We are always looking for an object, so then, numbers are words: ink marks on paper -- signs. And the question is: what gives those words meaning?
If 'two' is a word, why should it trouble us to call '2' a word? Numbers are number-words, or, mathematical-signs, i.e. that is the grammatical family they belong to, their part of speech. Numerals and operators ('+' and '=' e.g.) are not essential to mathematics; the English words 'two' and 'plus' would serve as just as well as '2' and '+' (although few people would master much math; cf. PG i, Appendix § 6, p. 223). And the equations or sentences of arithmetic would look like: 'Two plus two equals four.'
Mathematics is a language, or a collection of languages. A child would have every grammatical right to say 'I don't speak arithmetic'.
*
And what's that when it's at home?
Nothing would help heal our wounded understanding more than excluding the combination of words 'abstract object' from the language. Meanwhile we can always remember a question to be asked of anything unheard of, namely What's that when it's at home? That question turns our attention away from hazy pictures of ghosts in the ether or ideas in the mind -- towards the clearer notion of 'meaning' as 'rules for using linguistic signs', which is meaning we can objectively describe.
When philosophers use a word ... and try to grasp the essence of the thing, one must always ask oneself: is the word ever used this way in the language-game that is its original home? (PI § 116)
That is, we learned to use a sign in a language-game. And if we now divorce the sign from that game (i.e. "its original home"), we lose the surroundings that give the sign meaning -- i.e. the sign becomes undefined. (Z § 448)
[Cf. "Are numbers abstract or concrete?" We make a distinction between things that can be touched and things that cannot be, but that distinction is not universally applicable: not everything must grammatically be either A or ~A. Trying to apply that distinction to numbers is a category mistake: numbers are neither tangible nor intangible (cf. PI § 304: "We have only rejected the grammar that tries to force itself on us here"); those categories are not applicable to number.]
What's "number" when it's at home? A category (class) of words, such as the word 'three' (or '3') and the word 'five' (or '5'), with a use in the language, particularly in mathematics -- particularly in the language of mathematics (although if maths is to be called a "language", it is not a natural language).
Following rules and 'understanding'
According to Professor Hegel (ca. 1812):
The young pupils must lose their eyesight and their hearing, they must be diverted from thinking concretely, be withdrawn into the inner night of the soul. [Note 3]
This is our old story: -- abstract nouns are names of abstract objects, so that if we are to understand their meaning we must not picture anything concrete.
What do you suppose Hegel counted as a clearly presented lecture -- how do you know whether a blind and deaf student has understood you? A grammatical question about how the word 'understand' is to be applied in Hegel's circumstances.
The grammatical criterion for the students' understanding is: that they do not try to apply the grammar of object-words to their professor's abstract-words: in other words, that they have learned the rules of this game. What, if anything, goes on in the "inner night" of the students' souls does not enter it at all. Understanding is a public event.
Note 1: Another example. I only remember the rule that the result of multiplying two negative numbers together (i.e. their product) is a positive number. And therefore the question 'What is the square root of -1?' is a meaningless combination of words (an unknown language) for me. I do remember that a calculus (i.e. rules, definitions) exists in which the combination of words 'square root of -1' has a role (a meaning). -- But I do not remember that calculus, I do not make use of it -- and further it is not necessary for me to make use of it in order for me to use the calculi that I do use, e.g. addition and subtraction ("adding and take away") etc.
Do you think the combination of words 'square root of -1' must have a meaning, just because e.g. 'square root of 4' has? And if you think that -- then where do you think that meaning comes from? Do you think that "advanced mathematics" already existed before mathematicians invented it -- i.e. that maths was discovered rather than invented? (In the sense that new mathematical rules are required to be consistent with the rules already in use, you might say that. But on the other hand, do you think that the rules of badminton already existed before human beings invented that game, that the rules had a pre-existence or eternal existence like Plato's Forms? That is not a mathematical question; it belongs to the Philosophy of Mathematics.)
Is it not rather the case that human beings invented a meaning for the expression 'square root of -1'? What mathematicians invented was a game ("language-game", a set of rules or a protocol) where the question 'What is the square root of -1?' is a move in the "game".
Similarly, I know that there is a calculus in which division by zero is defined. -- However, I do not remember that calculus -- and I need not remember it. I might even be mistaken in thinking that such a calculus exists. My ignorance of that mathematics does not matter to the mathematics that I do know: because it does not have a role -- to use Wittgenstein's simile -- in the language-games that I do play.
Is a game you can play played wrongly if someone can invent additional rules for it to have? Are children not "playing maths" if they know nothing beyond addition and subtraction? Are two people not playing a game of tennis if they do not know the rules of the extension called "doubles tennis"? [BACK]
Note 2: The idea -- the apparent conclusion -- that there is no such thing as a mathematical object seems to appeal to no one.
A picture is conjured up which ... forces itself on us ... but does not help us out of the difficulty, which only begins here. (PI §§ 426, 425)
Plato's reality "on the other side of the sky" populated by shadows, namely numbers, even more hazy than "the absolute triangle". We clothe numbers with the same material we clothe spirits (PI § 36) -- "What is the material of an immaterial object?" With language we make absurd analogies which don't appear absurb to us. And this is because we don't understand the logic of our language, and until we find a way to make an objective distinction between sense and nonsense, that logic will remain misunderstood by us. This is the importance of Wittgenstein's work.
It may be a case of being held captive by a picture without being able to imagine an alternative picture, the picture being that "All nouns are names and the meaning of a name is the thing the name stands for", that "thing" being an object of some kind. (Of course, I have been using the word 'picture' here very loosely: is an unseeable picture a picture?) [BACK]
Note 3: Quoted by Franz Wiedmann in his Hegel: an illustrated biography, tr. Joachim Neugroschel (New York: 1968), p. 44. [BACK]
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とても興味深く読みました：
再生核研究所声明353（2017.2.2） ゼロ除算 記念日
2014.2.2 に 一般の方から100/0 の意味を問われていた頃、偶然に執筆中の論文原稿にそれがゼロとなっているのを発見した。直ぐに結果に驚いて友人にメールしたり、同僚に話した。それ以来、ちょうど３年、相当詳しい記録と経過が記録されている。重要なものは再生核研究所声明として英文と和文で公表されている。最初のものは
再生核研究所声明 148（2014.2.12）: 100/0=0, 0/0=0 － 割り算の考えを自然に拡張すると ― 神の意志
で、最新のは
Announcement 352 (2017.2.2): On the third birthday of the division by zero z/0=0
である。
アリストテレス、ブラーマグプタ、ニュートン、オイラー、アインシュタインなどが深く関与する ゼロ除算の神秘的な永い歴史上の発見であるから、その日をゼロ除算記念日として定めて、世界史を進化させる決意の日としたい。ゼロ除算は、ユークリッド幾何学の変更といわゆるリーマン球面の無限遠点の考え方の変更を求めている。― 実際、ゼロ除算の歴史は人類の闘争の歴史と共に 人類の愚かさの象徴であるとしている。
心すべき要点を纏めて置きたい。
１） ゼロの明確な発見と算術の確立者Brahmagupta (598 - 668 ?) は 既にそこで、0/0=0 と定義していたにも関わらず、言わば創業者の深い考察を理解できず、それは間違いであるとして、１３００年以上も間違いを繰り返してきた。
２） 予断と偏見、慣習、習慣、思い込み、権威に盲従する人間の精神の弱さ、愚かさを自戒したい。我々は何時もそのように囚われていて、虚像を見ていると 真智を愛する心を大事にして行きたい。絶えず、それは真かと 問うていかなければならない。
３） ピタゴラス派では 無理数の発見をしていたが、なんと、無理数の存在は自分たちの世界観に合わないからという理由で、― その発見は都合が悪いので ― 、弟子を処刑にしてしまったという。真智への愛より、面子、権力争い、勢力争い、利害が大事という人間の浅ましさの典型的な例である。
４） この辺は、２０００年以上も前に、既に世の聖人、賢人が諭されてきたのに いまだ人間は生物の本能レベルを越えておらず、愚かな世界史を続けている。人間が人間として生きる意義は 真智への愛にある と言える。
５） いわば創業者の偉大な精神が正確に、上手く伝えられず、ピタゴラス派のような対応をとっているのは、本末転倒で、そのようなことが世に溢れていると警戒していきたい。本来あるべきものが逆になっていて、社会をおかしくしている。
６） ゼロ除算の発見記念日に 繰り返し、人類の愚かさを反省して、明るい世界史を切り拓いて行きたい。
以 上
追記：
The division by zero is uniquely and reasonably determined as 1/0=0/0=z/0=0 in the natural extensions of fractions. We have to change our basic ideas for our space and world:
Division by Zero z/0 = 0 in Euclidean Spaces
Hiroshi Michiwaki, Hiroshi Okumura and Saburou Saitoh
International Journal of Mathematics and Computation Vol. 28(2017); Issue 1, 2017), 1-16.
http://www.scirp.org/journal/alamt http://dx.doi.org/10.4236/alamt.2016.62007
http://www.ijapm.org/show-63-504-1.html
http://www.diogenes.bg/ijam/contents/2014-27-2/9/9.pdf
http://www.ijapm.org/show-63-504-1.html
http://www.diogenes.bg/ijam/contents/2014-27-2/9/9.pdf
再生核研究所声明371（2017.6.27）ゼロ除算の講演― 国際会議 https://sites.google.com/site/sandrapinelas/icddea-2017 報告
http://ameblo.jp/syoshinoris/theme-10006253398.html
1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12276045402.html
1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12263708422.html
1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12272721615.html
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