Our mathematics appears not to be fixed in type (how we produce the end result), but perhaps is fixed in kind (the end result, our intuition rendered into logic). Its type depends on our choice of logic system, our choice of foundational axioms, and our choices for representation of mathematical objects.
Are There Other Systems Of Mathematics?
There are other possible choices for these variables, which produce different mathematics from the ‘classical’ mathematics above.
Constructivist mathematics is one such alternative. It begins with a similar idealistic (of the mind) view of mathematics as above, but ultimately restricts objects and logic to strict construction. The conceit is that only constructed objects have meaning, and mathematics without meaning is religion.
The constructivists prohibit axiomatic definitions and non-constructive proofs, enforcing a ‘build it or don’t talk about it’ discipline. In the process, they must switch to a modified axiom set and a modified logic. The core of idealistic mainstream mathematics has been re-developed successfully from this constructivist foundation. This is an example where the ‘kind’ of mathematics we intuit appears invariant under an axiom/logic transformation.
Idealists continue their construction project, bringing a useful pragmatism to the job. Mainstream mathematics and physics have a tradition of continuing to play the game even when the supporting understanding gets thin. Von Neumann remarked “In mathematics, you don’t understand things. You just get used to them.” And Feynman advised people who were protesting too much the lack of intuitive meaning of quantum mechanics: “Shut up and calculate”.
The late twentieth century has brought a new foundational contender, category theory, an outgrowth of homological algebra and algebraic geometry. Although most mathematicians seem content to continue working within ZFC, there are ongoing discussions of new foundations in category theoretic circles, for instance Elementary Theory of the Categories of Sets (ETCS). It is already projected that category theory has the power to reconfigure boundaries between mathematical disciplines, bridging logic, set theory, meta-mathematics, algebraic geometry, and computer science.
The future development of topics in these disciplines promises to look quite different from our current type of mathematics. To create a top-down umbrella with potential to unite all mathematical inquiry, category theory seems a likely place to devote effort. But here, we are describing a meta-mathematical layer, a language in which to express the axiomatic foundations and structural commonality across all mathematical endeavor. Should it come to pass, ‘absorb it and forget it’ will become even more cogent advice to the average mathematician working on concrete problems.
There have been many ‘schools’ of mathematical philosophy, dealing with what we know of mathematical objects and how we know it. Let’s say only that we know these abstracted objects by their mathematical definitions and by our intuition, which assigns them form and meaning. Mathematics is of our imagination, and it is through our imagination, our mind’s eye, that we know it. Perhaps to create is to know.
The workers at the frontiers of mathematical logic are putting mathematics on a firmer, more consistent, more abstract footing. The powerful tools they are developing may be necessary to crack open our most perplexing mathematical nuts. Yet there is perhaps danger, as they aim for the heart of what we know and how we can know it, that they will incidentally render mathematics lifeless, death by over-abstraction. For instance, ETCS, sometimes fondly called generalized abstract nonsense by its practitioners, may lead to bizarre mismatches in levels of sophistication when the generalized machinery is applied to a concrete problem, thus impacting beauty.
It is tough work out on the frontier. Language is inadequate to frame their questions. Their logic is challenged to avoid antinomies. For those asserting a real existence to mathematical objects, reality seems inadequate to frame their ontology. But the edifice of mainstream mathematics gains an ever surer footing through their efforts.
Mathematics may outgrow ZFC. It will be nice to have a new foundation ready in time, so our continuing construction project won’t miss a beat. We hope the revamped design remains pleasing to the mind’s eye.
How Much of Mathematics is Currently Known?
Some have suggested that mathematical knowledge is open-ended, that the edifice can be under construction in perpetuity. But being an invention of the human mind, one can contrarily suggest real limits. And ultimately, after all necessity has been provided for, the driving forces behind mathematical construction may wane.
Mathematics has matured dramatically since the ancients began the construction; the future for development seems more limited than 3000 years ago. For one thing, the majority of mathematics that will ever be accessible to the casual observer, say with a university degree, is already mostly known to us now, unless of course mathematics becomes completely reinvented. But there is no sign of that yet; what was true 3000 years ago is still true.
As the frontiers get pushed further out, the number of years of study needed to make an advancement keeps increasing, so that perhaps in the distant future, this novitiate period may approach the duration of a human productive lifetime. At that point, maybe we’ll have to say enough. For instance, 200 years ago, young mathematicians were making startling new discoveries by the age of 20. We now typically see such notable advancements by mathematicians in their mid- 30s and later. At this rate, mathematics runs out of steam in another 500 years or so. Remember, you heard it here first.