What is Algebraic K-Theory *really*, Grandpa?

Mathematicians, physicists, chemists, and hamster-fanciers  do not wonder how to define their subject (not until they have tenure, at any event), but simply roll up their sleeves and dig into it.

Later, in their rocker,  on the front porch,  bourbon on the sideboard and surrounded by the admiring upturned faces of their numerous grandchildren (always providing that such creatures shall still exist in years to come, rather than being all solipsistically oblivious  as, fondling their J-Pads or whatnot  with downturned personae and sightless eyes behind their encasing of Google-glass, they text their various virtual-avatar BFFs in unbreakable code, never stirring from the vat of amniotic fluid in which they float forever wihout sensation, all their internal organs having long since been harvested by their Reptilian Overlords for use in necromancy --

-- but I wander from my topic -- forgive it -- an effect of age),

later, as I began to say, when, fallen into the sere, the yellow leaf, and sicklied o’er with the pale cast of philosophic doubt, exposed to the bitter winds of (and so forth),

some of them may indeed eventually turn their minds, not to mah-jong, but toward certain Questions which, while not in any easily specifiable sense more “fundamental” than the deepest truths of their own chosen subject, are yet, in a precise sense, “meta” to them:  exempli gratiâ :  What are we really about, when we mathematize / botanize / admire our hamster?

Other than to those who have themselves delved deeply into such matters -- in particular, to those actively engaged in clearing new forests in their own fields, and with no time for such marginalia-- the answer may strike one as purely post-hoc, decorative, suitable for framing, worthy of being enshrined in a museum, but in no wise analogous to a fingerpost, or to fuel in your tank.   

Yet consider the following passage from a research mathematician and (for my generation) prominent textbook writer:

The main problem of topology is to decide when two spaces are homeomorphic.  From this point of view, the main problem of Operator Theory is to decide when two operators are unitarily equivalent.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963).

Here, the answer to “What is Topology?”  is not simply knitted into a sampler and tacked to the drawing-room wall, but used as an actual heuristic for finding your way in a different field.

Soberingly, he adds (aye, and has the wounds to prove it):

“But usually this is too hard.”

Still, that did not mean that the field of Operator Theory was stymied; similar questions in the topology of low-dimensional manifolds were “too hard” until the work of Thurston,  Perelman, etc.
 

~

All of which is simply by way of invitation to our  essay on the subject,

“What is Mathematics?”

where these matters are discussed in greater depth and detail.

[Note:  The quotation marks are largely ironic.]

~

Appendix: 

“What is Chemistry?”

Walter White, in the pilot episode of Breaking Bad, introducing the subject to his high school class:

Teacher: “Chemistry… is the study of … what.”

Pupil: “Chemicals.”

Teacher: “ ‘Chemicals’ -- N-n-n-n-NO!

Chemistry is: technically, the study of matter. 

But I prefer to see it as the study of:  change.”

 

(That  presages the very great change that he himself is about to under go -- a veritable transition to a new phase-state.)

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