Of course you always have to be careful with AI - it makes stuff up so needs confirmation — Gary Venter

I have adequate knowledge of two areas of thought and/or practice: mathematics and rock climbing. Yesterday I asked ChatGPT about a close friend of mine in the latter, what he is best known for. AI produced a reply that was entirely wrong, stating my friend was famous for a certain climb, while in fact he never did that climb and is known for an entirely different accomplishment. Made up facts.

A farewell to one of the few military veterans on the site, as if that means anything on a philosophy forum. But Wittgenstein was a military hero. Did that influence his writings?

Possibly. But I suspect from observations over time that, for example, women seem to have an ability to categorize and perceive relationships that correlates with algebraic structures.
And there is an ongoing discussion concerning men vs women in regard to spatial perceptions.

From ChatGPT:

some studies have indicated that women tend to be more represented in certain areas of mathematics compared to others. For example, fields such as mathematical biology, mathematical education, statistics, and applied mathematics have seen relatively higher levels of female participation compared to more traditionally male-dominated areas like pure mathematics or theoretical physics.

I might have guessed math education, but not the others.

When we try to understand reality only through external physical things imagined from this outside position, we lose sight of the necessity of experience. This is the Blind Spot, which the authors show lies behind our scientific conundrums about time and the origin of the universe, quantum physics, life, AI and the mind, consciousness, and Earth as a planetary system

I wonder how this might apply to math research?

Another QM interpretation holds that the quantum field is a Hilbert space, not just mathematically but actually, which would make the physical world part of the set-theory universe, reversing the question this thread raises. I personally find this non-appealing for a few reasons. — Gary Venter

From ChatGPT 3.5:

Quantum field theory (QFT) is a theoretical framework that combines quantum mechanics and special relativity to describe the behavior of elementary particles and their interactions. In the context of QFT, the quantum fields themselves are typically described mathematically as operator-valued fields defined on spacetime.

A Hilbert space is a mathematical concept used to describe the state space of a quantum system, where states are represented by vectors and physical observables are represented by operators. In quantum mechanics, the state space of a single particle is often described by a Hilbert space.

In quantum field theory, the state space becomes more complex due to the infinite degrees of freedom associated with fields defined at every point in spacetime. The state space of a quantum field theory is typically described by a Fock space, which is a direct sum of tensor products of Hilbert spaces associated with different numbers of particles. Each mode of the field (corresponding to a particular momentum) can be thought of as a harmonic oscillator, with its own associated Hilbert space.

So, while individual components of a quantum field theory can be described by Hilbert spaces, the full quantum field itself is typically not described by a single Hilbert space, but rather by a more complex structure known as a Fock space.

Maybe this will get the thread back on original tract. Thanks for your comments. I agree that Hilbert spaces are useful for the manipulations of Q Theory, but are more descriptive than fundamental.

:up:

But the point stands nonetheless — Lionino

Sorry, I missed this discussion. How does this bit about group theory relate to postmodernism?

Commenting on this, I don't think it is quite correct to say that multiplication is not defined in <ℝ,+>. It doesn't exist in the scope of <ℝ,+> of course, but whether it is defined or not is not a pertinent question. But the point stands nonetheless — Lionino

Postmodern?

Show me how you will determine the calculation of input values and a binary operator after you die; show me how the universe will determine the calculation of input values and a binary operator after all sentients die. — ucarr

Sorry. No idea what you are talking about.

The effect comes from the cause (by definition), so the effect includes the cause — LFranc

This has been my take on mathematics. Once certain definitions and processes are settled, all that logically flows from them does, in a sense, exist. It's a matter of discovery.

You’re saying it’s not empirical in the way that a novel or poem is not empirical, right? — Joshs

Sounds right.

math philosophers who argued for the construction of mathematics as an empirical endeavor. — L'éléphant

I think of Newton, developing calculus to describe physical phenomena. And perhaps some math is created in this fashion today. But by and large it's not an empirical process. Although math is called the Queen of the Sciences, it is not really a science.

↪jgill

My version — Mark Nyquist

That's OK. We all have our versions.

Absent any physical theory, logic says non-existent and non-physical things don't have any cause and effect relation. — Mark Nyquist

Is mathematics non-existent? Some might say yes. It's certainly non-physical. But cause and effect run thorough it like an intellectual river. In a typical theorem there is an hypothesis which gives rise to a conclusion.

- Brazillian Jiu jitsu;
- Drums, voice, guitar, bass, keys.. few others, including Irish Whistle!;
- Songwriting in light of the above - 23 albums and counting;
- Free Running/Parkour (mostly handstands and other power moves);
- Writing comedy for television and other stand-ups;
- Writing battle raps that will never see the light of day (though, there is footage of me doing several battles out there on the internet... )
- Collecting/enjoying Whisky/ey and fine Wine;
- Currently Learning Spanish and Arabic;
- Trying to solve the origins of the Voynich manuscript;
- Visiting puppy litters; and
- Writing science fiction (two pieces, thus far.. but one is a Trilogy for which i've only begun the first volume). — AmadeusD

Are you real, Amadeus, or are you an AI bot? :gasp:

AI can think like a human: it plagiarizes well.

Yeah those were the days — Sir2u

Look at my icon, from a BASIC program on infinite compositions of complex functions. :cool:

No, math is not embedded in the universe. There are structures and patterns we recognize and attempt to describe using mathematics, which we initially devised for this purpose. But mathematicians love to explore this thing we have devised, and see where it might go in various directions, regardless of whether it continues to describe physical reality. Hence, all those tens of thousands of research papers per year - explorations into the abstract.

even though the world has moved on, long covid still has me by the throat: — 180 Proof

Sorry to hear that. Hope progress in treatment comes about.

Writing mathematics programs in BASIC. Love that language. Physical exercise when possible.

I believe it makes perfect sense to say set x is only a member of itself in its own set — Philosopher19

No problem. If it makes sense to you, that's OK. Mathematicians don't have to agree with you. It's not like the fate of the world hinges upon this. It's OK to feel good about your own creation. Why argue with others?

:clap:

I consider the Butterfly effect as a kind of philosophy of time travel. In the movie The Sound of Thunder, a hapless time traveler in the age of dinosaurs steps off a designated artificial path and crushes a flower or something tiny. He then returns to his time and finds a different President and a city overgrown with vines and lions roaming the streets. In mathematical terms this is called SDIC, sensitive dependence on initial conditions, and is a feature of chaos theory.

But not much is made of situations that are not as sensitive. Stanislaw Lem had an Ergodic theory of time travel in which minor disturbances in the past are evened out over time and are insignificant.

Mathematical models of both these "philosophies" can be constructed, probably using a Banach space with vectors being "incidents". Time might be an operator with features we might not think of out of this context. For example if time contracted events even slightly Ergodic theory could prevail. The contractive element of time might stem from relativity, but probably not. A Banach space has certain conditions to be met.

Here is a path for philosophical conjecture that is more accessible than rambling endlessly in wormholes and loops.

There used to be a force of gravity. Now mass curves space and that produces what acts like a force, but explains the details better — Gary Venter

I think gravity is still a force, and mass curves spacetime. Its a little like mass increasing as velocity increases, a perspective replaced by increases in kinetic energy. Sorry. Nitpicking.

Those aspects of analytic philosophy that intersect mathematics, like set theory, are certainly more than idle chatter. And philosophies of ethics and law certainly are relevant. But arguments over what the great philosophers of the past meant seems more like speculative nostalgia than substance. As for the discussions on TPF , well, its a fun place to engage in prattle.

a little like MWI except alternate worlds (in Hilbert space) — noAxioms

Hilbert space? Don't end up there. You'll be a sentient vector upon which linear operators prey, trying to leave the premises but caught there by completeness, finally manhandled by an ominous inner product.

I don't think there is any hope for time travel.

It's so easy for practitioners of the subject:

$2(x+5)\equiv 2x+10$ Identity

$2(x+5)=3$ Conditional

I agree. There was developing an interesting discussion on the law of identity and (non-ordered) sets. Or so it seems, I just glanced at it.

Only if you skate much better than your peers. In the sport of climbing one must be truly outstanding and even then the money is not great. You might end up being sponsored with your monetary rewards in free roller skates.

I decided to determine if I had been a postmodernist mathematician, so I found an article on researchgate : The Proceedings of the 12th International Congress on Mathematical Education.

Math research is like a giant tree, with a more or less solid core, but with branches upon branches proliferating endlessly. There are so many of these no human can understand more than a fraction of the mathematics represented. So, in a sense, "mathematics" is ill-defined. Postmodernism pushes beyond this surmisal to the point of melting away the rigor of elementary mathematics, allowing the student to play with a subject they know little about, setting aside established principles and rote practices.

So any notion that math is a single connected body of knowledge is muted and an effort is made to disorganize what has barely been organized. Then there are DEI considerations, which may lead to practices that raise one's eyebrows if not their hackles, like ending the practice of grading and testing or manipulating advanced placement policies.

I retired from college teaching twenty four years ago having never been involved in these approaches, beyond being advised to be especially nice to minorities - which I had always practiced. So, it appears to me that PM mathematics is mostly a factor in mathematics education. I have never known or even met a research mathematician who considered themselves post modern. Guess I'm not either.

We see a lot of interest in
physics and the other sciences, but it seems new ideas in
maths are rare . . .What am I really saying? Is research mathematics just
a pointless pastime... — Christoff Montnielsensons

I should comment here. arXiv.org is a repository of mathematical research papers. Even then an author must be recommended by another author who has been approved by the same process.

The last time I looked about 5,200 articles were submitted in the 48 days of the new year. Over a hundred a day. So new ideas in math are anything but rare, since virtually all papers do indeed have new results and not hashing over previous ideas.

Looking at Logic (includes set theory and foundations) one sees perhaps five papers a day. History and Overview category is two or three a day. But when you read the titles and see the abstracts almost all fall under other categories, like the papers titled "Real analysis without uncountable sets" and "The double gamma function and Vladimir Alekseevsky". There is no category of "Philosophy of Mathematics" beyond these two. For philosophy of mathematics one must leave arXiv.org and go to PhilArchive, or a similar repository.

"Pointless pastime"? For the hundred thousand or so practitioners around the world it is a challenging exploration of ideas. For most others it is pointless if it doesn't provide support for physical projects.

Don't know that book, but

Ax x*0 = 0 is an axiom of first order PA, so it's easy to prove x*0 = 0 — TonesInDeepFreeze

Maybe it was aI=I. I don't recall. (In the recovery annex of the hospital recuperating from a broken leg at age 87) :sad:

I believe the solution to Russell's paradox is in here:
http://godisallthatmatters.com/2021/05/22/the-solution-to-russells-paradox-and-the-absurdity-of-more-than-one-infinity/ — Philosopher19

Honestly, I am having trouble dissecting the arguments used here.
Thoughts, jgill ? — Lionino

Russell's Paradox and infinity arguments hold no interest for me. After going round and round with the author on First Causes, I suspect I would learn little from this paper.

For me, as a kid, New Math was wonderful — TonesInDeepFreeze

That's great. Some kids really liked it, even though their parents didn't. I only tried teaching it in a typical college algebra course using a book by Vance. The first chapter was elementary set theory. My students had the ordinary curriculum in elementary through high school and for the most part were aghast at having to reason that a0=0.

I've just begun experimenting with AI systems, raising questions about factual issues that I have intimate knowledge of. I am appalled at the differences in answers.

I say the US education system does a massive disservice to the field of mathematics due to the fact that it divorces the philosophy of mathematics away from the applied version. — Vaskane

the philosophy of mathematics in the 20th century was characterized by a predominant interest in formal logic, set theory (both naive set theory and axiomatic set theory), and foundational issues.

In fact, this was attempted in the New Math of the 1957- 1970s. It was a disaster. For a variety of reasons. I know, I was there in the classroom.

Thanks for your thoughtful and intelligent reply.

The "math boys" here at the forum tend to respond with 'go read some math texts' to anyone who disagrees with them on fundamental principles — Metaphysician Undercover

You have mentioned, for example, that the limit concept is flawed, although it works well most of the time. But I don't recall your argument beyond that point. A more complete knowledge of space and time and points and continuity? Oh yes, something about the Fourier transform and the Uncertainty principle. What are your suggestions to fix that up? Intuitive mathematics? Remind me where doing something specific makes it better.

Are you working on a change in the fundamentals of math that might calm your concerns? I hope so, no one said math as it stands is perfect.

This thread is like a causal chain. What would you say about its first cause(s)?

Relevant: Lawvere's fixed point theorem. — Lionino

Good for you. I flamed out at "epimorphism". (i.e., the beginning). And I have actually worked with fixed points in Banach spaces and specifically the complex plane.