Kirwin Hampshire emits a cri de coeur in his The Dark Night of Mathematics. He comes across as slightly unhinged with such remarks as :
I am going insane. During the last week or so LLMs have produced a number of counterexamples to significant long-standing conjectures.
and
I am suffering a profound spiritual crisis due to these developments. I have been screaming internally for days. It feels as though I am living inside of a nightmare.
His piece is nonetheless worth reading. What caught my attention in particular was this observation:
Everything said thus far still sidesteps the emotional core of the issue. Here it is:
There is something about mathematical discovery (progress, advancement, creation) which is vital to the spiritual, experiential quality of doing mathematics. The creation (or even the pursuit) of novel mathematics is one way that humans have historically accessed the ineffable and encountered the divine and mystical.
That admission may come as a surprise to some non-mathematicians. But I would be willing to bet that for any mathematician reading this, what I have said above is quite prosaic—whether or not it accords with their personal experience of mathematics. Here is a brief gesture at the full sweep of mathematical mystics and dreamers: Ramanujan, Grothendieck, Cantor, Pascal, Luzin, Leibniz and possibly you, or someone you know.
I am not a mathematician, but I know some advanced mathematics and have studied set theory, and some of the papers of Georg Cantor. The spiritual thrust of Cantor’s work is undeniable and well-documented by Joseph Warren Dauben in Georg Cantor: His Mathematics and Philosophy of the Infinite (Harvard UP, 1979).
I am not a mathematician either, but I have read enough philosophy of science to o be dangerously free with my opinions about these matters without having to go through the hard work of actually learning science, statistics, or mathematics. It’s nice work if you can get it.
Mathematics is atemporal, like logic, the Platonic Ideas, and God. We non-mathematicians rarely appreciate this, because the mathematics we encounter is a structure of the world that is useful for everything from business profitability to the orbits of the planets. For us, then, mathematics is grounded in temporal time, structurally dictating how things are or will behave.
But the mathematician enters into the mysteries of math in its pure atemporality and is confronted with the unique timelessness of its truths. If you define the objective as that which abides whether we are consciously aware of it or not, then the truths of mathematics qualify. 2 + 2 = 4 is always true within mathematics’ self-contained logic; the shortest distance between two points is always a straight line in two dimensional Euclidian geometry. Learning such truths is a contingent event, but what is learned is not: they are objectively true in every time and circumstance.
Epistemologically, mathematical objects are unique as well. In temporality, there is a gap that separates thinking about being from being itself (the existence/essence distinction). But in mathematics, thought and being are identical; the algorithms define the objects exactly. To grasp the logic is to apprehend the complete truth about the objects, without remainder or possibility of error.
All of this implicates a realm beyond the time-bound particulars of the world we know, and can stir a yearning for the ineffable. As Werner Heisenberg, the father of quantum physics, put it: “The first gulp from the glass of natural sciences will turn you into an atheist, but at the bottom of the glass God is waiting for you.”
Tom,
You didn’t address Kirwin Hampshire’s *cri de coeur,* but given my Platonic sympathies, I agree with most of what you say. The truths of arithmetic, for example, are true at every time, at every place, and in every possible world.
You say “But in mathematics, thought and being are identical . . .” Consider the number 7 (not the numeral ‘7’). 7 exists necessarily, i.e., in every possible world, including worlds in which there are no thinkers and thus no thoughts. Unless you bring God into the picture, it follows that thought and being are not identical.
I think what you want to say is that mathematical objects are wholly intelligible unlike objects of sense perception which are not
>>An Algorithm is a set of step-by-step instructions for solving a problem or completing a task, similar to a recipe. In computing, algorithms tell a computer how to process inputs such as data to produce an output, such as a decision, prediction, or sorted list. Some algorithms are simple and fixed, while others “learn” patterns from data, as in machine learning.<<
Are there all the algorithms there could have been in Plato's *topos ouranios*?
Thanks for the comments. You provide food for thought as usual. I am not sure I have much to say about Hampshire’s cri de coeur; perhaps I will post a comment on that if I come up with anything worthwhile.
But on to a couple of your points.
“I think what you want to say is that mathematical objects are wholly intelligible unlike objects of sense perception which are not”
I agree that mathematical objects are wholly intelligible. My formulation was in contrast to the duality of being and thought implicated in the existence/essence distinction. But my point remains: as wholly intelligible, a mathematical object has no being apart from its intelligibility; it is identical with the logical thought that describes it. And yet, the object does not depend on anyone actually thinking it. As you point out, the number 7 necessarily exists whether or not it is thought. This is marvelous in a way that objects of sense perception are not. And I submit that it helps explain why mathematicians often acquire a sense of the ineffable.
“Are there all the algorithms there could have been in Plato’s *topos ouranios*?”
Algorithm, as used on social media, is little more than a synonym for “solution to any problem.” It invokes the aura of a deep thought coupled with mathematical rigor without supplying either. As you note, it is also used to describe technical processes like those in computer programs. Given the term’s colloquial and technical usage, my choice of that term may have been distracting. Nevertheless, I used it in the restricted sense of a rigorous proof or demonstration of a mathematical object, which would encompass the wholly intelligible nature of a mathematical object.
With that in mind, I’ll take a stab at your question. Short answer, no. The algorithms of logically self-complete and certain mathematical objects might sit comfortably in Plato’s Heaven. But the application of such objects to the contingent diversity of the phenomenal world would require algorithms that are at best conditional probabilities. Such uncertainty is not the true knowledge reserved for Plato’s Heaven. As Einstein said, “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” – Geometry and Experience, January 27, 1921.
Bill
The author of the post you referred to has expressed a genuine feeling of a mathematician at AI situation. One of my mathematician friends feels much strongly about “AI systems stealing real mathematician’s work”. Imagine how software developers feel…
On a side note I am not certain I understand well the “mathematical objects are wholly intelligible”. I know that this sentence is an expression of a Platonic sentiment in the philosophy of mathematics. But some mathematical objects are so weird that at least some contextual constraints are in order in that sentence. What I mean here is that math objects without context or frame of reference could hardly be called wholly intelligible. Like looking for 1.5 in the set of natural numbers. Or making sense of 2+2 = 1 (makes sense in base 3 not in decimal) or Weierstrass function. I am not trying to be cute. Just expressing a thought (Nelson Goodman has a great influence on my thinking as you know)
Dmitri,
Your first paragraph makes an important point glossed over by ‘rah rah’ AI proponents such as Trump and Mark Levin. It’s going to put a lot of people out of work. Take novelists. There is a big market for formulaic schlock ‘beach novels.’ AI engines could easily crank out stuff as good as the schlocksters . . . And why do I need an M. D. to interpret the results of a CT scan? And so on.
As for your second para, take the number 7, not the Arabic numeral ‘7.’ One context here is the infinite set of the naturals, which I take to be actually and not merely potentially infinite. If there is this object 7, then there have to be aleph-nought of them. So yes context matters.
‘7’ is notation used in base-10. But one and the same number can also be represented in binary notation and in base 3 notation. Would you agree? Not if you are a nominalist, of course, which I think you are. I grant that there is something we could call ‘representational’ context. We can discuss this further if you like. Time for dinner and the news. Curious how Netanyahu was received at the U. N.