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I am reading “The Nature of the Physical World” by Sir A.S. Eddington.
The Nature of the Physical World
Written in 1928, AFAIK it predates any category theory or thinking, what with MacLane still two years away from graduating Yale at this point.
Yet as I read it, I can’t help but get the impression Sir Eddington was already thinking categorically: he speaks of a world-building in terms of relations only, speaks of space and time as categories, clearly states the limits of set theory and laments that there are no other systems; in fact, MacLane’s “adjoints are everywhere” quip adequately describes the underlying philosophy of “The nature of the physical world” .
Given the prominence that Eddington had at that time, I find it unlikely that MacLane was not exposed to his writings and, as I’m asking, I actually believe it might have had a great influence on his categorical thinking.
Is there any reference or literature to this effect? I’ve not read any biographies of either men, so maybe this connection is well represented in those works. The most modern mathematical treatment of Eddington’s ideas that I could find is this 2004 PhD Thesis, but it doesn’t touch on category theory.
Sir Arthur Eddignton and the foundations of physics
For those that have read it, am I just suffering from categorical pareidolia?
I know of nothing that remotely suggests this.
Eddington is not mentioned in Mac Lane’s autobiography, or indeed in any of Mac Lane’s writings that I know of. You don’t mention Eilenberg for some reason, but I don’t know of any Eilenberg-Eddington connection either.
Keep in mind that Eilenberg and Mac Lane invented categories and functors in order to provide a conceptual niche in which to define the notion of natural transformation. This was an act of hard conceptual analysis, and was originally motivated by technical considerations in algebraic topology. Here’s an extended quote from Barr & Wells, Toposes, Triples and Theories:
Categories, functors and natural transformations were invented by S. Eilenberg and S. Mac Lane (announced in “The general theory of natural equivalences” [1945]) in order to describe the connecting homomorphism and the long exact sequence in Čech homology and cohomology. The problem was this: homology was defined in the first instance in terms of a cover. If the cover is simple, that is if every non-empty intersection of a finite subset of the cover is a contractible space (as actually happens with the open star cover of a triangulated space), then that homology in terms of the cover is the homology of the space and that is the end of the matter. What is done in Čech theory, in the absence of a simple cover, is to form the direct limit of the homology groups over the set of all covers directed by refinement. This works fine for defining the groups but gives no information on how to define maps induced by, say, the inclusion of a subspace, not to mention the connecting homomorphism. What is missing is the information that homology is natural with respect to refinements of covers as well as to maps of spaces. Fortunately, the required condition was essentially obvious and led directly to the notion of natural transformation. Only, in order to define natural transformation, one first had to define functor and in order to do that, categories.
I suppose I’m biased by reading the book (MacLane) and not the paper (Eilenberg & MacLane) but more importantly because MacLane graduated Yale in physics in 1930.
This is roughly 10 years after Eddington made a rock-star out of Einstein while being his most eloquent evangelist. And thus expose to Eddington’s words and views would have been as likely as his exposure to Einstein. And as I read “Nature of the physical world”, I can’t help but get a sense that Eddington was already thinking along the lines of categories of objects and the natural transformations between them.
I could cite several examples from the text to support this point but I’m not saying “Nature” is correct or relevant or a precursor in any way to any form of category theory . I’m not even saying Eddington was striving for the same things as Eilenberg and MacLane; after the former is a physicist seeking plausibility while the latter are mathematicians seeking proof.
I’m just observing on what I think may be an example of proto-categorical thinking to give it a name: A physicist struggling, reaching for a tool that would be invented by mathematicians a year too late (Eddington died in 1944).
But thank you for confirming that there is no historical account of Eddington influencing the pioneers of category theory; many thanks!
but more importantly because MacLane graduated Yale in physics in 1930.
That’s not accurately put. He double-majored in physics and mathematics in his junior and senior years. And it’s clear that he was much more interested in mathematics than in physics by the time he graduated. (He reports that he was unaware what exciting things were happening in physics in those days; I assume he means the quantum mechanics developed in the 1920’s.) So to say he “graduated Yale in physics” is, I think, misleading.
The main takeaway for Mac Lane from physics was Hamiltonian mechanics, to which he would return later as a mathematician. I suggest reading his interview in More Mathematical People to get a fuller picture.
I can’t help but get a sense that Eddington was already thinking along the lines of categories of objects and the natural transformations between them.
I imagine you will have a hard time convincing others of the part about “natural transformations” (in the mathematical sense) in Eddington.
Occasionally you will hear claims that someone or other had contemplated the rudiments of category theory before E & M. I’ve heard this about Ulam, Brandt of groupoid fame, and maybe Mackey or Stone as well. But having known Mac Lane pretty well, I am certain that he would reject, in strongly worded language, the suggestion that someone else planted such suggestions. He and Eilenberg say they did see category theory as a continuation of Klein’s Erlanger Program. (If he were alive and heard you propose that Eddington influenced him, he’d probably really let you have it. (-: )
But having known Mac Lane pretty well, I am certain that he would reject, in strongly worded language, the suggestion that someone else planted such suggestions.
My question is in the spirit of “inspiration”, not directly “planting suggestions”; the apple indirectly inspired newton, it didn’t plant a suggestion. Likewise the original query, now rejected, that Eddington may have inspired MacLane somehow via his physics background.
(If he were alive and heard you propose that Eddington influenced him, he’d probably really let you have it. (-: )
If he were alive, I would welcome the oppurtunity. :)
I imagine you will have a hard time convincing others of the part about “natural transformations” (in the mathematical sense)
It is undoubtedly a matter of historical record that E&M get credit for proving nat. trans; this by itself doesn’t prove they were the only people thinking along those lines. Especially people in physics, as my original query asserts, may have felt the necessity of a nat. tran. but lacked the machinery to implement or even conceive of what form it should take.
I am not seeking alternative venues for that proof nor to imply that E&M’s ideas were not their own; I’m merely examining the idea that the plausibility of categorical thinking was already present long before E&M formalized it, as I called it proto-categorical thinking:
” We take as building material relations and relata. The relata are the meeting points of the relations. The one is unthinkable apart from the other. I do not think that a more general starting-point of structure could be conceived.” p. 116
“We often think that when we have completed our study of one we know all about two, because “two” is “one and one.” We forget that we still have to make a study of “and. Secondary physics is the study of “and” ⎯that is to say, of organization.” - p. 52
” But the space of physics ought not to be dominated by this creation of the dawning mind of an enterprising ape. Space is not necessarily like this conception; it is like—whatever we find from experiment it is like. Now the features of space which we discover by experiment are extensions, i.e. lengths and distances. So space is like a network of distances.” - p. 41
shrug
This discussion (which I don’t think I’ll add more to past this comment) and the quotes you adduce remind me a little of those books by Fritjof Capra and Gary Zukov which claim that 20th century developments in quantum field theory and general relativity were somehow already implicit or foreshadowed in Buddhist or Vedanta philosophy. The idea of networks of relata (categories!) could similarly be seen as elaborating on the idea of Indra’s Net and the interdependence and interpenetration of all things.
So were the Yogic and Taoist masters also engaging in proto-categorical thinking? Yes, maybe? Generally these speculative analogies seem idle to me, and not very fruitful.
Generally these speculative analogies seem idle to me, and not very fruitful.
As a mathematician seeking proof and not plausibility, I would expect no less of you and I sincerely thank you for engaging me thus far and sharing your insights.
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