Wittgenstein Tractatus 6.23-6.3432 on a priori intuitive logic about the world, supported by calculation and mechanics
With sister Hermine on not-fun maths, L's Tractatus error, the illogicalities of naturalistic artistic imagination, and avoiding bust-ups over busts and breasts
Ludwig Wittgenstein wrote:
[MGH: the art of logic]
Tractatus 6.23-6.3432
6.23:
If two expressions are connected by the sign of equality, this means that they can be substituted for one another. But whether this is the case must show itself in the two expressions themselves.
It characterizes the logical form of two expressions, that they can bę substituted for one another.
6.231:
It is a property of affirmation that it can be conceived as double denial.
It is a property of “1+1+1+1” that it can be conceived as “(1+1)+(1+1)”.
6.232:
Frege says that these expressions have the same meaning but different senses.
But what is essential about equation is that it is not necessary in order to show that both expressions, which are connected by the sign of equality, have the same meaning: for this can be perceived from the two expressions themselves.
6.2321:
And, that the propositions of mathematics can be proved means nothing else than that their correctness can be seen without our having to compare what they express with the facts as regards correctness.
6.2322:
The identity of the meaning of two expressions cannot be asserted. For in order to be able to assert anything about their meaning, I must know their meaning, and if I know their meaning, I know whether they mean the same or something different.”
6.2323:
The equation characterizes only the standpoint from which I consider the two expressions, that is to say the standpoint of their equality of meaning.
6.233:
To the question whether we need intuition for the solution of mathematical problems it must be answered that language itself here supplies the necessary intuition.
6.2331:
The process of calculation brings about just this intuition.
Calculation is not an experiment.
6.234:
Mathematics is a method of logic.
6.2341:
The essential of mathematical method is working with equations. On this method depends the fact that every proposition of mathematics must be self-evident.
6.24:
The method by which mathematics arrives at its equations is the method of substitution.
For equations express the substitutability of two expressions, and we proceed from a number of equations to new equations, replacing expressions by others in accordance with the equations.
6.241:
Thus the proof of the proposition 2 × 2 = 4 runs:
(Ων)μ’x = Ων×μ’x Def. — Ω2×2’x = (Ω2)2’x = (Ω2)1+1’x = Ω2’Ω2’x = Ω1+1’Ω1+1’x = (Ω’Ω)’(Ω’Ω)’x = Ω’Ω’Ω’Ω’x = Ω1+1+1+1’x = Ω4’x.
6.3:
Logical research means the investigation of all regularity. And outside logic all is accident.
6.31:
The so-called law of induction cannot in any case be a logical law, for it is obviously a significant proposition.—And therefore it cannot be a law a priori either.
6.32:
The law of causality is not a law but the form of a law.
6.321:
“Law of Causality” is a class name. And as in mechanics there are, for instance, minimum-laws, such as that of least action, so in physics there are causal laws, laws of the causality form.
6.3211:
Men had indeed an idea that there must be a “law of least action”, before they knew exactly how it ran. (Here, as always, the a priori certain proves to be something purely logical.)
6.33:
We do not believe a priori in a law of conservation, but we know a priori the possibility of a logical form.
6.34:
All propositions, such as the law of causation, the law of continuity in nature, the law of least expenditure in nature, etc. etc., all these are a priori intuitions of possible forms of the propositions of science. …
6.343:
Mechanics is an attempt to construct according to a single plan all true propositions which we need for the description of the world. ….
6.3431:
Through their whole logical apparatus the physical laws still speak of the objects of the world.
6.3432:
We must not forget that the description of the world by mechanics is always quite general. There is, for example, never any mention of particular material points in it, but always only of some points or other.
[MGH: the logic of art]
Caption family photo 1917: “Kurt, Paul, Hermine, Max, Leopoldine, Helene, Ludwig”
LETTERS CONCERNING MATHEMATICS AND ART
[replies are not provided in the source material]
FEATURING: Michael Drobil (1877–1958) academic sculptor and member of the Wiener Secession, close friend made by Wittgenstein while prisoners of war.]
… and Drobil’s bust of Ludwig, commissioned by Hermine with misgivings …
Bust of Wittgenstein by Michael Drobil
Hermine Wittgenstein to Ludwig Wittgenstein
5 November 1920
My good Lukas,
You can imagine that it wasn’t a matter of indifference to me to be stuck up here instead of being able to be with you; I cursed, and continue to curse, because the prospects of my being able to visit you in the foreseeable future are not good. I have to take it so pitiably easy in a way hitherto unimaginable to me, and whenever I start to think ‘oh, what, the doctor’s an ass; I’m going to do it differently this time’ I learn right away that I’m the one who’s the ass, and not the doctor. Of course it would be different, if I were alone in this world or if it were of any import to the world that I perform some definite activity at precisely this moment; but as things stand I have to see to it that I make a full recovery, though I shudder at the thought. I wonder how long it’ll take. Mima wrote me an extremely kind letter yesterday and described Trattenbach and everything; I can picture it perfectly already. I was very sorry, however, that she was at Drobil’s place and spoke to him the way she did; but everything she writes is precisely what I myself thought.
From the heads in the Secession, I at once saw that Drobil’s aim and ability consist in looking for and illustrating the multitude of minor forms in nature: i.e., in a certain sense, in exaggerating. This profusion of minor forms is characteristic of the heads of women and small children, and his search has been rewarded quite nicely. But whenever he looks for such profusion on a head that seems to have been worked over by a carpenter (of course they exist there too, but to such a negligible degree that their apparent absence is what’s characteristic), whenever he exaggerates these minor forms – and he must exaggerate them, if he truly wants to express them – he acts against what’s characteristic of this particular head and that exacts its revenge. For a portrait must be characteristic, if it’s meant as a portrait. I’m just glad I wasn’t there; I certainly wouldn’t have been able to hold my tongue and this is exactly a case in which anything one says is really like trying to tell an apple tree that it ought to grow pears! It of course does not want to grow pears, and it is still an apple tree even if pears are ten times more what’s required. I am writing to you in such detail about this because you’re incapable of imagining that Drobil can be so far off the mark despite all his good qualities.”
—At the moment I’m carrying out an experiment that puts me to shame: mathematics was always difficult for me as a child, though I later thought my teachers were mostly to blame for that because they never taught me how to do it properly. Now I’ve brought a 1st and 2nd grade maths book here with me because I would like to be able to do basic school maths for a certain purpose. And I often sit here for hours at a time with rules and exercises that a ten-year old is expected to understand before I really take them in. No one watching would be able to understand it at all; he’d be bound to think that I’m distracted and that my thoughts are wandering all over the place, when really I’m concentrating on proportions, calculating fractions, etc. The following proposition can be found in your book: ‘The processes of calculating serve to bring about that intuition. Calculation is not an experiment’: [Tractatus 6.2331]. That’s not true when I do calculations; they’re always experiments, and I’m always very anxious to find out what the result will be. I now understand that I was unable to understand it as a child; but I still don’t understand how any child can understand it!
Keep well, my good Lukas,
warmest regards,
your sister, Mining
Hermine to Ludwig
1 March 1921
I was recently at Drobil’s studio and was very glad to find that the bust is much better than I expected it to be. It is of course a ‘Drobil bust’ insofar as it displays his soft spot and love for hidden forms, but it also displays such a likeness that I was able to praise it honestly and from the heart; I hope he noticed. I dislike his other works so much, especially, e.g., his wife as a maenad with a body marred with traces of all the dismalness of the war and post-war years; it’s just too much of a failure for my tastes and I wasn’t able to find words for it. I believe portraits in the broad sense of the term, which can also be understood as portraits of a landscape or of a still life, are the only thing artists are still capable of doing today. Even a Drobil, it seems to me, can’t muster up the imagination to sublimate elements of reality; rather, he stops short at love and loyalty.
Hermine to Ludwig
9 October 1922
Drobil was here on Sunday and started chipping away at the unsightly Adam’s apple on that a bust of you. Every splinter that fell away was an act of charity, and it became clear that the base has to be as small as possible, but just big enough for the bust to stand on its own. Unfortunately, the tools were blunted rather quickly, so that we can only continue working on it next Sunday; I would have really liked to have seen it done already.
Ludwig to Hermine
14 January 1925
Dear Mining!
Thank you for your letter. As long as Drobil hasn’t ruined anything by enlarging the breast!!!
The pronounced concave breast was necessary. The reason is it’s very possible that some fundamental nonsense has happened. Because the breasts mustn’t form four uniform hills together with the upper arms, which would give the overall impression of a corrugated inclined plane forming a background to the whole.
Nor should the space between the right lower arm (the vertical one) and the breast be any smaller, as that space will otherwise become insignificant and – so to speak – accidental. Please convey my reservations to Drobil; he’ll understand them (if he wants to!). That the breast wanted fixing is true, but that it wasn’t to be fixed by enlarging the bosom is more than probable, and Drobil himself, as you will remember, told me in front of you that he was not going to take the breast out any farther than he already had.
The matter is – I believe – not at all so simple. He ought to think about it for a few weeks rather than just cooking up something superficial that is basically nonsense!
Auf Wiedersehen.
Your brother,
Ludwig
The Sources:
Ludwig Wittgenstein, Tractatus Logico-Philosophicus, translation by C. K. Ogden and F. P. Ramsey, Kegan Paul (London), 1922
Wittgenstein’s Family Letters, edited by Brian McGuinness, Bloomsbury Academic, 2019
Evolutions of social order from the earliest humans to the present day and future machine age.