I was recently impressed by a series of tweets from Joscha Bach. I’ve never heard somebody use this language before.
That’s a wonderful and provocative notion that accords with my own quixotic ideas. Perhaps we don’t need to reform mathematics, but replace it altogether. That sounds like fun.
What’s wrong with math?
If you ask a computer scientist, they might explain how there’s a large gap between theoretical math and applied math—mathematicians claim they can do things which computers cannot.
If you ask a physicist, they might tell you about the large gap between math and physics—many objects and processes in mathematics cannot exist in the real world, even in principle.
But if you ask me, I see this as an outcome of a deeper problem: the philosophy of mathematics is a total mess and has been for more than a century.
I recently came across a document I had written many years ago about this subject for a friend. Instead of an article, it’s a bunch of bullet points. I thought it did a rather nice job of summarization, so I’ve shared some of the points here.
The Situation with the Philosophy of Mathematics in the 20th Century
- The foundational crises that happened around the turn of the 20th century have not been resolved correctly.
- Some mixture of the three dominant schools—logicism, intuitionism, and formalism—is likely correct.
- From logicism: mathematical truth is grounded in logic.
- From intuitionism: mathematics is a human language which is designed to capture our own concepts.
- From formalism: we must allow the utility of mathematics to stand apart from its truthfulness; bad math (even conceptually incoherent math) is often useful.
- Some mixture of the three dominant schools—logicism, intuitionism, and formalism—is likely correct.
- The metaphysics of mathematics has not been sorted out properly.
- Mathematics does not speak for itself, and it often does lie.
- The meaning and purpose of mathematical axioms has to be clarified.
- Godel’s incompleteness theorems are overrated in their significance, and the proof might be entirely sidestepped with alternative mathematical philosophies.
- His axiomatic framework was explicitly formalist.
- The proof is baroque and hard to follow.
- Who said Godel numbering is a legitimate mathematical process?
- Georg Cantor’s idea of the transfinite was logically incoherent.
- He ultimately justified his ideas with appeals to God and Divine Revelation
- “One proof is based on the notion of God. First, from the highest perfection of God, we infer the possibility of the creation of the transfinite, then, from his all-grace and splendor, we infer the necessity that the creation of the transfinite in fact has happened.”
- He ultimately justified his ideas with appeals to God and Divine Revelation
- The concepts of infinite totality and infinite sets are logically flawed and must be extricated from the foundations of mathematics.
- Historically speaking, these concepts are new and few mathematicians from previous centuries would have accepted them.
- The concept of continuity is therefore logically flawed and needs to be replaced with the concept of absolute discreteness—that all appearances of continuity are actually underpinned by discrete processes.
- Computers have finally proven that we don’t need the concept of mathematical continuity anymore. All continuity is discrete continuity—in other words, the appearance of smoothness is generated by underlying discrete processes.
- We need discrete, logical replacements for so-called irrational, real, and transcendental numbers, and for the concepts of convergence and limits.
- Therefore, the formal theory of calculus will need to be re-worked.
- There will likely be a discrete translation key to rescue these concepts—e.g. computers are already able to do mathematics without utilizing any of the aforementioned concepts.
- I expect we’ll find specific examples in computer science, though I don’t think whether we’ll find a general theory or general language yet, which is the ultimate goal.
I know there are others with similar intuitions. I’ve spoken with some of them on my podcast. I expect 20th century orthodoxies to be replaced within the next few decades. The momentum is too great on our side; computers have been too successful. The gap between “theoretical math” and “applied math” is too large to ignore.
If we can’t reform, replace!
