Procedure-based (as opposed to definition-based) concepts

Question

Euler's work on divergent series was guided by computational procedures, rather than any definition of the "value" of such a series. E.g., he was happy to have half a dozen procedures that indicated that $0!-1!+2!-3!+\dots$ equals 0.59637..., even though he had no definition of what an expression like $0!-1!+2!-3!+\dots$ means.

This style of mathematics is not very common nowadays; some purists would say you're not even doing math if you can't give a definition of the objects you're studying. But I don't think many people would disparage the work of Renaissance algebraists who studied complex numbers without benefit of the modern concept of complex numbers as ordered pairs.

What are some good modern examples of mathematical research in which one is guided by computational procedures that seem to give mutually consistent results, rather than by definitions of what the procedures are purportedly computing? (I suspect that quantities that physicists computed in the late 20th century using ad hoc regularization techniques would be one class of examples.)

Answer 1

Having mentioned Euler and twentieth century physicists in the question, Pierre Cartier's 'Mathemagics? (A Tribute to L. Euler and R. Feynman)' should provide good cases for you.

My thesis is: there is another way of doing mathematics, equally successful, and the two methods should supplement each other and not fight.

This other way bears various names: symbolic method, operational calculus, operator theory . . . Euler was the first to use such methods in his extensive study of infinite series, convergent as well as divergent. The calculus of differences was developed by G. Boole around 1860 in a symbolic way, then Heaviside created his own symbolic calculus to deal with systems of differential equations in electric circuitry. But the modern master was R. Feynman who used his diagrams, his disentangling of operators, his path integrals . . . The method consists in stretching the formulas to their extreme consequences, resorting to some internal feeling of coherence and harmony.

Presumably you already know of John Baez The Mysteries of Counting: Euler Characteristic versus Homotopy Cardinality

Answer 2

The idea of $\mathbb{F}_1$, the field with one element, might fit the spirit of the question. (If you feel it does not, please, let me know and I will remove the contribution.)

The idea that there should be a mathematical object resembling a finite field with a single element was put forward by Tits in the 1950s. Having a satisfactory theory of the 'field with one element' might allow progress on major mathematical problems (in number theory in particular), as it could allow to adapt the proofs of results, such as the Riemann-Hypothesis, known in the 'geometric case' (e.g., curves over finite fields), to the 'arithmetic case' (e.g., integers).

Various investigations related to this were and are undertaken. One could thus say there is mathematics on a concept without it being defined. For a recent contribution see for example Fun with $\mathbb{F}_1$ by Alain Connes, Caterina Consani, Matilde Marcolli.

Answer 3

Turing machines, recursive functions, the lambda calculus, cellular automata, register machines all seem to compute the same things but there's no overall definition of "computation". The idea that there is such a thing as "computation" derives from the agreement of these apparently disparate definitions.

Answer 4

There is a theory of asymptotic series. I guess you know that, but the usual perturbation series in quantum field theory do not converge. For example the scattering amplitudes for QED in their series representation given by the Feynman rules are known to be asymptotic series (and (perturbative) QED is not “late 20th century”, but about 60 years old.

I would not say that this is not very common: QFT and string theory always have to deal with such series. You might say that this is what physicists do, but not mathematicians. But in dealing with such series there are hard mathematical problems involved, that are considered by mathematicians. But modern mathematics are of course always “definition based” (well, sometimes even in mathematics the “right definitions” have to be found, but it is not like in theoretical physics where people are often not even seriously looking for the definition)—you define series as formal objects, which you can manipulate although they do not converge.

Answer 5

Newton iteration to find the zero of a function. I think, in general iteration processes lack usually "static" definitions; for another instance there is also this iterative procedure for testing Mersenne-numbers for primality named after Lucas and D. Lehmer. (There surely might "static definitions" be possible, but the overwhelming application of that two concepts should be the iterative/procedural way if I do not have a very distorted impression)

Answer 6

Not sure if what I will now say is correct at all, but still want to say it.

Maybe in fact at least in the area of foundations the procedure-based, as you call it, approach is inevitable. Because of incompleteness already when dealing with Peano Arithmetic, and much more so in Set Theory, we are in a position of approximating our object of study (like the structure of all natural numbers with certain amount of usual operations and induction principles or, say, some cumulative hierarchy of sets) by means of gradually adding new axioms and studying their consequences, never knowing at each given step whether we have added something too strong and reached contradiction. We know that we are doomed to do it again and again, never reaching the complete description of the structure in question.

I would say this kind of approximation process is a combination of the axiomatic method with the procedures that you mention, and presence of the latter seems unavoidable.