Does Math Have An End?

This article is part of a series on Intuitive Math Epistemology. See also:
1. Is Math Discovered or Created? 2. Does Math Have An End? 3. Tchaikovsky and Debussy

Is there a limit to the number of “true facts” contained in the discipline of mathematics? If (for the sake of argument) humans were around indefinitely, would the discipline ever end? Would we, one day, proclaim that we’d “reached the bottom”? Or does the field continue indefinitely into the depth, reaching arbitrary levels of complexity? If this is the case, where do these true facts come from, and why do they exist?

gravity

What will be the math of the future? What will math look like, a hundred years from now, or five thousand, or (again hypothesizing that we’ll be around indefinitely) even a hundred thousand?

These questions make math seem like a strange, magical and universal discipline. The pursuit of their answers has long eluded me. Here, I finally make some attempts.

Previously, we investigated how we make our way through the thick universe of mathematics. Is our path predestined, lighted by logic itself, waiting, only, to be discovered in the dark? Or do we ourselves light this wilderness, illuminating it with blazing color and creativity? We decided something intermediate: that though this light’s pathway may be predetermined, its color and feeling are determined alone by humanity.

Now, we ask a different question: where does this path lead? Does it have an end, and what could this end look like?

Proliferation of Axioms

We first investigate a basic question: whether arbitrary axioms indefinitely produce logical corollaries. In other words, given a single, particular set of starting principles, or axioms, is there an infinite supply of “true” facts which will logically follow from this set of axioms? This fundamental question will be important in answering our larger one.

I’m tempted to say that the answer is no. Each system of axioms produces a “space of consequences”, we might imagine, a universe of ideas which depend on these axioms. Upon reflection, however, it seems quite difficult to argue that this “space of ideas” could be infinite! Sure, this space may be massive, and very complex. The light has been shined into a “deep cave”, so to speak. But to argue endlessness would be to claim that these axioms trigger ideas, which then trigger more ideas, and so on, cascading indefinitely in an infinite chain reaction. From whence could the “material” for this continuation come? No cave – and nothing in our known universe, for that matter – is infinite. Ideas could in principle be infinite, probing endlessly through logic itself. But this would demand the existence of forces beyond those which I can explain.

Examples seem to support my hypothesis. Point-set topology depends on a miniscule, and highly abstract, list of axioms. The field has produced quite deep results, and remains extremely fundamental throughout all of mathematics. It’s largely considered, however, to be all-but solved. Group theory, too, rests on an elegant and extremely compact collection of axioms. Mathematicians, however, recently completed a hundred-year-old effort to classify every possible finite simple group: every possible direct ramification of these axioms (1). The cave was massive – the classification “theorem” spans tens of thousands of pages in hundreds of journal articles by hundreds of authors – but it was by no means infinite. The flashlight illuminates distant walls.

Coherently Building

We’ve established that a particular set of axioms might be exhausted. What about math in general? Here, the picture seems much more subtle. Before, only logic itself extended outwards from the axiomatic core. Now, however, humans build on this very core itself, adjoining new structures, new properties, and new relations. These additions must logically cooperate with their predecessors. Other than this, however, the room for expansion seems unlimited.

Previously, we asked “to what extent must something logically follow.” Now, we ask “to what extent could something logically follow.” These questions are very different. Their answers seem different, too. Indeed, new structures can always be defined. New structures constantly indeed are defined, and these structures – in the utterly awe-inspiring world of “modern mathematics” – reach complexity unimaginably staggering.

Each new augmentation might produce only a finite extension. But the number of augmentations could be infinite.

Interests and Transitions

Will it be? Here, we must ask a very different question: will humans continue to find reason to accumulate augmentations? This is perhaps the hardest question of all. Our task has changed from philosophical epistemology to predicting the future!

The answer, thankfully, seems to be yes. Math, we must remember, has endless (and often mysterious) applications – in physics, computing, biology and more. Much more importantly, however, mathematicians explore this world for its own sake. They explore the mathematical universe because it’s interesting. The magnificence of this universe will only continue to explode.

In the truly long run? It’s hard to tell. We’re over our heads! Perhaps math will be phased out in favor of a new discipline altogether – super-ultra-logic, say. Perhaps things will change in a still-more-unpredictable ways. We can’t know.

Math’s warriors, however, push forward undeterred. I’ll continue to do the same. Onward I charge into the black depths. Onward! Onward into the impossible abyss!

  1. Classification of finite simple groups
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3 comments on “Does Math Have An End?

  1. Josh says:

    Shinichi Mochizuki’s work seems to be the math of the future, almost by definition, since as of now no one can understand it. Either that, or his self-ascribed title of “Inter-Universal Geometer” might be worth taking less than lightly.

    In all seriousness, though, I think it’s worth asking whether the theoretical study of primes is one channeled by our interests, or one that transcends them. I’d lean towards the latter. If anything were to fit the classification of “super-ultra-logic,” that would be the study of primes.

  2. Mattson says:

    It seems as though Kurt Godel has answered a portion of this question in that there will be some logical corollaries produced from a sufficiently strong set of axioms that, as of now, we (maybe just me) don’t really know what to make of. Godel’s incompleteness theorem (which you probably already know, but I will restate) says:

    1. In any consistent, and sufficiently strong, formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F.

    and

    2. Such a formal system cannot prove that the system itself is consistent.

    This was produced when Godel proved, using typographical number theory, his statement “G” and “not G”. As the formal system becomes sufficiently strong, it in essence destroys its own ability to be consistent by producing both a statement and its negation. Something akin to Heisenberg’s Uncertainty principle, applied to the strength of a formal system vs. its completeness as opposed to position vs. momentum of a particle, emerges. Know one too well, and the other becomes blurry.

    I think that math’s “end” will be the end of this math being “the” math. I think that the next steps are going to be working with and understanding things like G and realizing that maybe, while we do not experience paradoxes directly (we are not alive and dead at the same time, as far as we know), they are a fundamental part of the reality in which we reside.

    And about your “super logic”, I agree. Perhaps in the same way that we now embrace the imaginary plane in physics to explain real phenomena, we will embrace some kind of “irrational” logic to explain real answers in mathematics.

  3. Richard says:

    We’ve discussed some of these issues before, Ben, so I won’t say too much.

    1. It would have been helpful to distinguish between ‘math’ in the sense the body of knowledge and practice of mathematicians and ‘math’ the various truths/facts which obtain/exist and are properly described as ‘mathematical truths/facts’.

    2. It would also have been helpful to distinguish math being infinite from math having no end. Because there could be infinitely many mathematical truths pertaining to a single branch without there being an infinite number of distinct branches of mathematical knowledge.

    In the former sense in 1, math is only without end if the study of math is without end and this seems impossible to answer on empirical grounds.

    In the latter sense, in 1, of course, math is infinite. If the existence of mathematical truths/facts is mind independent then the axiom of infinity alone guarantees enough distinct mathematical entities for there to be infinitely many mathematical facts/truths.

    The fact is that, in doing mathematics, mathematicians for various conventional reasons mark out different parts of the vast mathematical structure differently. Presumably this has some epistemic utility to it, reflecting the exact cognitive capacities of human beings, but whether such demarcations will continue indefinitely is, again, currently impossible to answer on empirical grounds. At least, that’s how it seems to me.

    The sense in which Godel provided any answers here seems very unclear to me. The first incompleteness theorem might be interpreted as telling us that mathematical practice won’t exhaust mathematical truth, insofar as the former is carried out in the manner of formal theory building. There’s a reasonable amount of scholarly evidence to suggest that this was something Godel himself believed after achieving his results. In fact, even adding the unprovable statements as axioms doesn’t suffice, because of the “essential” incompleteness results which Godel gave.

    But the language of mathematics is only semi-formal, in the required sense, it’s basically a supplemented natural language, so I don’t see how exactly the Godel theorems apply to mathematics in general, for the purposes of this conversation. Also, Godel, and others, did/do feel they knew/know what to make of Godel sentences: they’re true, after all, just not amenable to formal proof within the systems for which they are the Godel sentences.

    It’s an abuse of Godel’s theorem to say that it is like Heisenberg’s Uncertainty Principle. Godel, after all didn’t think the Godel Sentences of a system were a mark of uncertainty, they were merely a mark of the limits of certain means of proof. The truths, if they are truths, expressed by Godel sentences can be known, he thought, and it is this which places the practice of mathematics (and abstract human thought generally) outside the scope of purely mechanical simulation.

    There’s a lot of debate over this though. Wittgenstein, notoriously (though it’s seldom mentioned) rejected Godel’s proof as establishing what it purported to establish. This was because he had arguments supposed to show that certain limits, like the limits of proof apparently revealed in the Godel theorems, do not necessarily indicate fundamental truths about reality so much as the limits of our own capacities. Godel’s arguments are designed to show that a system which has a certain property undermines itself by proving itself inconsistent. The Wittgensteinian stuff is subtle though so I won’t go into detail here.

    As to revision of Logic, there’s not really enough space here for me to say anything worth saying. The matter is fairly complicated when treated fully.

    Interesting topic though. Nice transmission of passion at the end.

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