p 36

Bohr and Wittgenstein?

Bohr’s observation ``We never know what a word means exactly’’ is quoted by Heisenberg: Wasn’t there any relation between Bohr’s philosophy and Wittgenstein’s philosophy (of his final period)? They shared very keen observations and critical attitude toward language.

To supply a tool of precise definition is an important function of a formal system (footnote 6)

J  Hintikka,  The Principles of Mathematics Revisited (Cambridge, UP, 1996) may beunderstood without pain if one have read something like H. D. Ebbinghaus, J. Flum, and W. Thomas,  Mathematical Logic (Springer, Undergraduate Texts in Mathematics, 1984). Excerpts from this book is posted at books: mathematics and physics . The conceptual analysis in this book is closely related the philosophy Hintikka stresses on p9 of his book as follows:

More generally, much of the foundational work that has been done since Cauchy by working mathematicians consisted in expressing in first-order logical terms the precise contents of different mathematical concepts.

 In developing such ideas [like ϵ-δ, integral], mathematicians were not engaged in the discovery of new mathematical truths. They were engaged in analyzing different mathematical concepts in logical terms.

 I will call this function of logic (logical concepts) in expressing the content of mathematical propositions its  descriptive function. If mathematical propositions were not expressed in terms of logical concepts, their inferential relationships would not be possible to handle by means of logic. 

 What I have called the descriptive function of logic can be put into service as a tool of

conceptual analysis. 

On Clarity

We must always carefully reflect on whether what we cannot speak clearly at present is really what we can never speak clearly or what we need more efforts to do so. Furthermore, we must not forget that there is no clear definition of `clarity’. This is parallel to the impossibility of rigorous definition of `rigor’ (M. Kline,  Mathematics the loss of certainty (Oxford UP, 1980) p315).

 There are many levels of `clarity.’ The level of clarity of mathematical logic is obviously different from that of the ordinary mathematics. The former clarity is, so to speak, to aim at avoiding any misunderstanding even by aliens who are quite different from us (or by computers with a primitive human-machine interface), but the clarity of the ordinary mathematics aims at `clear understanding’ by ordinary human beings. Still, the human beings assumed are not the `total human beings as social organisms’ but those whose capability is restricted and who is only with meager social experiences. The math clarity will not use the conspecific characteristics as  Homo sapiens positively . Chinese philosophers (Hundred Schools of Thought) or the Bible do not play with pedantic logic relying on the very nature of natural language. Still they have universally appealing power, because the readers share the same universe with the writers; we are both organisms on the earth. The use of language consciously appealing to the shared background with the readers is called `poems’ (or more generally `literature’). After all, using language consciously implies writing poems or doing mathematics.

 Those who place importance on languages must know how incomplete tools languages are.

Dyson on Fermi

I learned more from Fermi in 20 minutes than I learned from Oppenheimer in 20 years.  In 1952 I thought I had a good theory of strong interactions. I had organized an army of Cornell students and postdocs to do calculations of meson-proton scattering with the new theory. Our calculations agree pretty well with the cross-sections that Fermi was then measuring with the Chicago cyclotron. So I proudly traveled from Ithaca to Chicago to show him our results. Fermi was polite and friendly but was not impressed. He said,

“There are two ways to do calculations. The first way, which I prefer, is to have a clear physical picture. The second is to have a rigorous mathematical formalism. You have neither.” That was the end of our conversation and of our theory. [Freeman Dyson, “A Conservative Revolutionary,”  Talk at the banquet of the C N Yang retirement symposium, May 21-2, 1999, Mod Phys Lett A 14, 1455 (1999).]