Stupid Mail

It being winter and needing distraction from politics, from time to time, I've been watching some of the Youtubes in the sister channel of The Periodic Table of Videos, the Numberphile.  Which is fun, though sometimes the arguments they go through tax my ability to follow them in a video format.  A lot of that is better read than heard. 

Listening to one of them a while back I suddenly wondered if mathematicians would ever be so cluelessly arrogant to claim that they knew more than a tiny fraction of what there was to know about numbers, if there are any counterparts in the far more precise and exacting science of mathematics to physicists who claim that they're on the cusp of having a universal theory of everything to do with the physical universe, the kind of guy like Stephen Hawking or Sean Carroll who get loads of attention for such claims which, when you look at them with even a modest degree of skepticism, it turns out they're claims are pretty silly.  

I was thinking about the question that Carroll wanted to ignore the obvious answer to, that physics, cosmology has no theory of everything about even a single electron in the universe, there is not even a single electron that physics can know comprehensively, exhaustively, completely, there is not a single electron that physics can know everything about.  It can't even know everything they can think of to want to know about it at any given second if its existence, as noted here, that was one of the most important discoveries of physics in the early 20th century, the time of those whose shoulders Hawking and Carroll rest on. It's like those discoveries of the inability of science to know anything completely don't matter to them in their quest for glory.  

Anyway, I would guess that there isn't a mathematician who would claim that mathematics knows anything about any number, even one, even zero that is anything like a comprehensive view of the properties and characteristics of the number are.  Mathematicians tend to deal with philosophical issues at a far deeper and more sophisticated level than those in the physical sciences - they have to.  I would guess that every single advance in pure mathematics adds to the complexity of every number, certainly every one relevant to the argument in that advance.  I would guess that those implications include the number one, though I'd probably have to ask a number of mathematicians who might or might not agree with that idea.  I can't think of how any advance in pure mathematics wouldn't add to the complexity of every number which would have to be included in its enclosed universe of relationships.   And physical objects are far less knowable than mathematical objects.  There are few, if any, statements of physics which have the same level of proof that is typical of mathematics.  Eddington, in his Swathmore Lecture of 1929 noted:

The physicist who inveighs against the lack of coherence and the indefiniteness of theological theories, will probably speak not much less harshly of the theories of biology and psychology. They also fail to come up to his standard of methodology. On the other side of him stands an even superior being – the pure mathematician – who has no high opinion of the methods of deduction used in physics, and does not hide his disapproval of the laxity of what is accepted as proof in physical science. And yet somehow knowledge grows in all of these branches. Wherever a way opens we are impelled to seek by the only methods that can be devised for that particular opening, not over-rating the security of our finding, but conscious that in this activity of mind we are obeying the light that is in our nature.

I think Eddington was being way too optimistic in his idealistic view of the people who do science, their modesty in making claims for the scope of what they claim is not generally in evidence.  Absurdly bold and unsupported claims are more typical, especially as the nature of what is claimed becomes more obviously tenuous and speculative, whether that be the, at best, quasi-science of psychology or the absurdly ambitious claims of theoretical cosmology, especially those which abandon anything but the pretense of being based on observation of nature.  They are more like the worst of theologians who make similar bold and unevidenced claims than they are a mathematician who has to back up his claims with tight arguments and who never can forget that they base their work on axioms which cannot be proven.  The best of theology is both rigorous in its methods and humble in its claims.  Though as those who slam theology almost never have read much if any of it, they wouldn't know that. 

I will note that Eddington is reputed to have spent a good part of his last years trying to come up with his own Fundamental Theory which is, today, generally considered to have been a breakdown in his previous habits of skeptical judgement about his assumptions.  

You got that, Bunky?   No, I didn't think you would.