That's an interesting question. It refers I think to Platonism in the Philosophy of Mathematics. When one accepts the ideas existence independent of the "observer", then, it is discovery. When Mathematical objects do not exist without the action of the mathematician, then it's construction.
Perhaps some advertising of ny own work is allowed. If consistency is a hallmark of existence-as in some views on mathematical objects- then concrete mathematical incompleteness destroys this view.
J.F. Geurdes, Riemann Zeta function on the real axis, Lobachevskii J of Mathematics, 46(3), 1266-1270, 2025.
I think that this discovery of inconsistency in mathematics points in the direction of "Mathematical objects are discovered."
I've long liked the quote due to Leopold Kronecker: "God made the integers, all else is the work of man." The natural numbers seem natural (and they lead to the integers and rationals), but the reals seem more of an abstraction we distill from experience. I suspect the mathematical regularity we see comes from the laws of physics. That's the "Platonic realm" physical reality arises from. The abstract Platonic realm we discover (I like the word "recognize") is our abstraction of those physical laws.
I wanted to work in the similar quote by James Jean - “God is a pure mathematician!” - but I couldn't find the place. Recognising is a good word for it. I had a professor back in college who used to talk about recognising a page from God's book whenever one saw an elegant proof in physics. There is definitely something there, even if it is very hard to explain exactly what it is.
I did come across your article when I was researching this one! I read a lot of Penrose's books a while back, and it was nice to see a good discussion of his "three-world" model.
Thanks! I'm planning a follow-up post to it one of these days. I want to explore the putative contents of the P.R., especially with regard to the idea that notions of justice or beauty may reside there. I need to research some results from LLM research that suggest a "moral axis" in the data, and that seems to suggest a potential for moral absolutes. If such do exist Platonically, that also points in that direction.
Mathematics is the language used to discuss what we discover. As in the case of all languages, we learn by nurture. As in the case of all arts (Music, Painting, Writing...), some Mathematicians are driven by their nature to be masters and composers of new symphonies. Mathematics enables humanity to perceive, understand, and appreciate our "ZeitOrtGeist" in the universe.
That's an interesting question. It refers I think to Platonism in the Philosophy of Mathematics. When one accepts the ideas existence independent of the "observer", then, it is discovery. When Mathematical objects do not exist without the action of the mathematician, then it's construction.
Perhaps some advertising of ny own work is allowed. If consistency is a hallmark of existence-as in some views on mathematical objects- then concrete mathematical incompleteness destroys this view.
J.F. Geurdes, Riemann Zeta function on the real axis, Lobachevskii J of Mathematics, 46(3), 1266-1270, 2025.
I think that this discovery of inconsistency in mathematics points in the direction of "Mathematical objects are discovered."
Thank you.
Taking this moment to thank you for the wonderful job you do here in this space.
Thank you, it is always nice to hear from you (and other readers too of course!)
Useful tool but not the structure of what is observed.
I've long liked the quote due to Leopold Kronecker: "God made the integers, all else is the work of man." The natural numbers seem natural (and they lead to the integers and rationals), but the reals seem more of an abstraction we distill from experience. I suspect the mathematical regularity we see comes from the laws of physics. That's the "Platonic realm" physical reality arises from. The abstract Platonic realm we discover (I like the word "recognize") is our abstraction of those physical laws.
FWIW, I wrote about this just last month:
https://logosconcarne.substack.com/p/where-is-the-platonic-realm
I wanted to work in the similar quote by James Jean - “God is a pure mathematician!” - but I couldn't find the place. Recognising is a good word for it. I had a professor back in college who used to talk about recognising a page from God's book whenever one saw an elegant proof in physics. There is definitely something there, even if it is very hard to explain exactly what it is.
I did come across your article when I was researching this one! I read a lot of Penrose's books a while back, and it was nice to see a good discussion of his "three-world" model.
Thanks! I'm planning a follow-up post to it one of these days. I want to explore the putative contents of the P.R., especially with regard to the idea that notions of justice or beauty may reside there. I need to research some results from LLM research that suggest a "moral axis" in the data, and that seems to suggest a potential for moral absolutes. If such do exist Platonically, that also points in that direction.
Mathematics is the language used to discuss what we discover. As in the case of all languages, we learn by nurture. As in the case of all arts (Music, Painting, Writing...), some Mathematicians are driven by their nature to be masters and composers of new symphonies. Mathematics enables humanity to perceive, understand, and appreciate our "ZeitOrtGeist" in the universe.
"it would be shocking to encounter aliens speaking English, but much less surprising if they told us the digits of pi."
It would be a bit of a surprise if the digits were for base 10. Base 2 would be less surprising: 11.0010010000111111...