Being perceived is not what it is for something to exist — Banno

A breathe of fresh air. A history over time exists whether it is recorded through human perception or not. Paleontologists discover this truth frequently.

For those who suspect math underpins the character of nature, then the passage of time might well be understood in mathematical rather than philosophical discourse. What does the limit concept say about time? In the ever expanding galaxy of mathematical subjects does time arise?

This is news to me. Hillsdale is a conservative college frequently cited in the more conservative news shows, and I have no problem with that. However most K12 schools are public, charter or non-charter, and where I live we hear little about Christian schools, although they exist here.

The boom of which you speak is the mouse's squeak, but loud, not weak. :cool:

↪jgill
What do you think about TREE(3)? — Arcane Sandwich

I think nothing of TREE(3). From my background in classical complex analysis it is merely notation without any connection to my area of interests. Just another of more than 30K math entries on Wikipedia.

One: It has multiple meanings. One such meaning is: It means "1".
Two: It means "2".
Three: It means "3". — Arcane Sandwich

Now I see why you seem infatuated with Graham's Number. I looked it up and found that it's the largest number associated with an actual mathematical problem (in Ramsey Theory). I also read of Knuth Arrows. New to me and a universe away from my mathematical interests. But whatever spins your wheels.

I see from your location on your info page you have followed Yogi's advice and have taken the fork in the road. Sorry to see the only moderator who is a mathematician leave that position. Fair sailing.

The problem with Time dilation is that it is another hypotheses i.e. possibility if you could fly in the speed of light. Could you fly in the speed of light? Could anyone? Even if you did, the result is not confirmed. It is a hypotheses — Corvus

https://en.wikipedia.org/wiki/Experimental_testing_of_time_dilation

I see nothing of substance in this philosophical discussion of time. But, if something can be physically manipulated and scientifically measured, I wager it exists. Time dilation does just that.

Do we think that DOGE will go after enormously expensive health care spending, which first and foremost is expensive because corporations make profit from it? — ssu

Interesting comment. I've wondered about our health care system, and this past year I have discovered how well it functions for senior citizens during the treatments of a broken leg and cancer at age 87, with complications. It's been one year since the fall shattering my right femur, then, in hospital, finding I had cancer elsewhere. I have a Medicare Advantage plan provided by my public employee's retirement program, and pay into it monthly, but my out of pocket charges were virtually minute.

A good friend recently had triple bypass surgery, and his surgeon told him that at the age of 82 had he lived in Ireland he would not have had surgery and would have been sent home to die.

More the dimensionality of thought than geometry. You might illustrate by describing an educational curriculum in elementary physics embodying your concepts.

:up:

It's very simple to show that infinite sets are not atl the same size — Janus

A misuse of the word "size".

Category theory would be the philosophers companion here, but uh... we haven't been trained in category theory in school or in the university. That is really something lacking! — ssu

This has come up before. There are categories in my own subject of complex analysis, but in order to work with them you need a solid background of complex analysis at the beginning grad level. Now set theory can start literally at the bottom and work up. I've mentioned before my intro to the Peano axioms and beginning at 0 and ending (at the end of the course) with exponential functions.

Category theory seems to be more a graduate school offering, whereas set theory can be presented at a much lower level. However, "New Math" of the 1960s and 1970s flopped when this was tried. Feynman was very critical of the effort.

. . . so one could describe the situation like mathematicians have outsourced the philosophical problem to set theorists — ssu

I like this. However, category theory - which includes categories of sets - an outgrowth of algebraic topology and what ever else of similar abstraction seem to have gotten into the game.

I was fortunate that the large state university I chose to get my PhD over half a century ago had a perfectly adequate but not elite math faculty, and I was able to do my research in a subject arising from classical complex analysis. Had I been confronted with category theory or a similar abstract topic I probably would have switched to computer science or electrical engineering.

Complex analysis, itself, has apparently moved up the inevitable steps of abstraction to the point that the arXive.org collections of papers on the subject are unreadable to me.

Classical mathematics doesn't need "infinity" as a sort of number with associated features. The limit concept works well. However, modern math defines the term(s) and goes into abstracta.

From the OP article:

My position is first that mathematics is an exercise in pure logic. It is not a human construct.

I'm not sure how to interpret this statement. If math is an exercise in pure logic and pure logic is a human construct, then so is math. So, is pure logic not a human construct?

Thanks for your reply. Very informative.

3) Philosophy is the Goddess of the Sciences — Arcane Sandwich

I checked with Copilot and AI agrees with you. (Well, it used to be - it hasn't provided much guidance in recent years, unless one means scientists who speculate)

That's the "level of dignity" that Foundations of Mathematics has. Now whose "fault" is that? Do professional mathematicians need to take the blame here, yes or no? — Arcane Sandwich

Only those relative few who have an interest in Foundations. :roll:

I have two apples. But I want to eat three — Arcane Sandwich

You apply Banach-Tarski to one apple, turning it into two the same size, then eat all three. But only if you have faith in the Axiom of Choice. If you do not you might raise the question on The Philosophy Forum. Abundant deep answers are found there.

"Years and Years" (2019) A British limited series that explores what might have arisen during and after Trump's first term. Excellent cast and intriguing story. Alternate history. HBO.

↪Corvus
What about Combinatorics, Group theory, Set theory, Boolean algebra etc.?
The world is exactly the way these disciplines describe. — EnPassant

So, the world has transfinite ordinal numbers. Or does it?

:clap:

This happens in mathematics as well. Long involved arguments get turned over to grad students and end up disappearing. Make it short and concise.

Numbers are fictions, and no fictions have causal efficacy. — Arcane Sandwich

Numbers don't exist as fictions, they exist as brain processes — Arcane Sandwich

So, numbers are fictions that don't exist as fictions. Does The Maltese Falcon exist as fiction?

Word games

Draw a circle on the X, Y axis with radius pi. All points on the circumference except 4 of them are irrational numbers. No others are rational, — EnPassant

Assuming you mean the ordered pairs of real numbers that identify points on the circumference have at least one member, x or y, irrational, what are those four points? x^2+y^2=pi^2.

Mathematics is the consideration of the properties of magnitude and multitude in the absence of any other properties — Count Timothy von Icarus

Hmmm . . . never thought of it that way.

Mazur's article on category theory introduces one to modern mathematics. Not necessarily mathematics as practiced by a great many professionals. As time passes levels of abstraction increase and the subject seems more and more like philosophy and less and less like calculus, for instance.

I'm suspicious of a process whereby students end up as variations of their professors. — Tom Storm

Sounds like a typical PhD program in mathematics. Occasionally a student flies off in an interesting exploration of their own. Usually knowledge advances incrementally.

This is logic's job. It is not limited to mathematics because mathematics is only one of the many formal domains of study, and the way that mathematicians progress in knowledge will not be identical to the way that other specialists progress in knowledge within their own field. — Leontiskos

This gives pause for thought. When I think of progress in math compared to progress in physics, say, the initial step of A to B requires speculation and experimentation, or with math, finding examples. In other words, one must posit the B. After that the mathematician works to logically connect A to B, while the physicist is perhaps more interested with finding counterexamples that would invalidate their theory.

Set theory is closely tied to logic, and lies in an area overlapping mathematics and philosophy. So one might ask whether Philosophy of Mathematics is a part of philosophy. I say it is, but others might disagree.

The notion that scientific laws and maths are contingent human artifacts rather than the product of some Platonic realm seems more intuitively correct to me. — Tom Storm

I agree.

But as an untheorized amateur, I would say that. — Tom Storm

We are all amateurs in this regard. Mathematicians rarely spend their time discussing or arguing the issue. It has so little to do with traditional math research. But times change for any human endeavor, and "modern" mathematics, category theory e.g., is an elevated and abstract perception of the way the subject has been for millennia and is possibly closer to the Platonic conundrum, although I can't see how.

And...does that mean I can't trust anything science says? — Darkneos

Of course not. Science works pretty damn well. If you were an astronaut would you distrust the science that got you to the moon and back? The proof is in the pudding.

So why is it that mathematical predictions so often anticipate unexpected empirical discoveries? He doesn’t attempt to explain why that is so, as much as just point it out. — Wayfarer

Indeed. My late ex-father-in-law, a Hungarian aristocrat - exchanged letters with Wigner, and he translated a few of these for me. I don't find it surprising that occasionally a development in math portends a scientific discovery. Mathematics arose from observing phenomena in the physical world, and those initial discoveries generated logical consequences, some of which provide insight into that same physical realm.

Because once eaten they are no longer "in themselves" but in us? — Janus

Mathematics is a very social endeavor. We explore, discover or create the subject, then place our results on paper, then digitalize and submit for others in the profession to read. Initially, it is a product of our minds, then when others read or hear about it, it becomes part of their minds, as well. If they find it interesting they may pursue the topic further and the process repeats itself.

Frequently, I can draw lots of images with pen and paper that help me understand a math idea. That helps make the topic "real". But there are limits. I can draw a line, an interval, and a point (you know what I mean), but I cannot draw an infinitesimal, no matter how tiny a point I can scratch on the paper. For me, infinitesimals are the metaphysics of math. Something I can work with but not develop an intimacy. I leave that to math people who indulge in non-standard analysis (NSA) or hyperreals.

Incidentally, there are very few universities that offer more than a course or two or independent study in NSA. The only two in the USA seem to be U of N Colorado, and U of Wisconsin - even there the pickings are slim.

Best not to be captivated by infinitesimals. The limit concept came along and did the little buggers in.

Are you saying divisibility cannot be "divided up" and/or sets displaying "evenness" cannot be divided up? For example, the set of even numbers can be divided up into those even numbers having exactly two 2s.

Humans seem to have evolved to the point of both constructing and exploring mathematics. The counting numbers arise from observations and abilities to distinguish. In my opinion none of math exists in some Platonic realm independent of human brains. These are ideas, not physical objects.

Modern math is concerned more with overseeing the multitude of mathematical ideas and discovering how they relate to one another, than the classical approach to conjuring up problems to solve in the individual areas of the subject (there are about 30,000 Wikipedia pages on math topics, e.g.)

On the other hand, I can't say these ongoing philosophical arguments are of less importance than much of the math being produced. My own areas of exploration are "pure" mathematics and have little to any connections to physical realities. Nevertheless, the results are documented in simple exercises of logic on a set of symbols that are well defined. I don't get the impression that is the case in Platonics. But I could be wrong.

If abstractions are mental content that's different and it should be acknowledged. And the infinitesimal as mental content is one possibility out of many. — Mark Nyquist

Good luck. Beware of serious babble on this thread. :roll:

Perhaps 1 is necessary, but not sufficient for 2. Just babble from my perspective.

This IS the mistake we do.

We START from natural numbers as it's the natural place to start for counting. It basically a necessity for our situational awereness, hence even animals can have a rudimentary simple "math"-system. Yet simply as mathematics has objects that are not countrable, starting with infinity, infinite sequences and infinitesimals, whole math simply cannot be based on natural numbers. This is the reason why Russell's logicism faced paradoxes. Not everything was discovered. That there exist the uncountable should make it obvious to us that natural numbers and counting isn't the logical ground on which everything mathematical is based upon. — ssu

Of course, "whole math" is not "based" on natural numbers. But they did come first. It was a start, like a path of a thousand miles, one step at a time. Those simple initial steps may culminate with climbing a thousand meter peak. Get a grip, man.

But numbers, and other ‘objects of reason’, are real in a different way to sense objects. And that is a stumbling block for a culture in which things are said to either exist or not. There is no conceptual space for different modes of reality (leaving aside dry, academic modal metaphysics). Which is why we can only think of them as kinds of objects, which they’re actually not. They’re really closer to kinds of acts. — Wayfarer

Well said. Starting with the natural numbers, which are ways to distinguish objects and converse about quantities, mathematics has grown to virtually unimaginable proportions over the millennia. And it has changed character from a descriptive and predictive tool to an enormous game, unbounded in some aspects, with recently formulated foundational rules.

Some compare it to chess, where material pieces are moved around a board rather than the pen or pencil upon paper, or keys and screen of a computer. Where it might differ is in potential: mathematics awaiting discovery or creation versus possible strategies or moves on the chessboard. Chess players might comment on this.

Is a crossword puzzle real? Pondering how to fill in the spaces, then doing so with pencil. Sounds a little like math. Are emerging ideas real? Of course they are. Do mathematical objects exist in some exotic realm, awaiting discovery? I think of them as commonalities of minds, the way in which human brains have evolved.

Do infinitesimals exist (in the platonistic sense)? — Michael

I've always thought of these little critters as part of the metaphysics of mathematics. They now belong to a variation of the game called nonstandard analysis.

↪jgill
Plato suggested momentary collapse — magritte

Would you elaborate, please?

Action at a distance might be momentary separations of time from spacetime. If space only exists all things are connected instantly.

Now this is posited as an alternative to the Labour idea of giving each household a sum in order to offset the cost of electricity. — Banno

When the government gives money directly to citizens for a particular purpose that money may seem like an invitation to celebrate in various ways rather than use for its intended purpose. That happened here under covid, although there was no pretense it be used for children's health, etc. The temptation to celebrate might not be as strong if gas prices are reduced a bit. Just a thought.

1) Both came to power rather by accident; — Linkey

Yeltsin groomed Putin for his political ascension. Then resigned. No accident.