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.

I find it interesting that the PSR and P of Causality seem to overlap regarding mathematical causality chains. Clearly sufficient reason and sufficient cause are there, whereas reason is more logic and cause is more physical. Just thought I'd go back to the OP.

Under certain conditions, time separates from spacetime. Or not.

Potential is the intersection of space and time. Or perhaps a synonym of an instant in time. Just rambling.

Were the phenomena nefarious it would not have the blinking colored lights similar to aircraft.
Unless they are meant to be seen and the magnitude of observations and commentary studied for national security.

Or, mostly private airplanes or helicopters.

If something doesn't exist, it is not possible to write about it. If something is being written about, then it must exist somewhere. — RussellA

You mean all the science fiction books are real stories? Or merely exist in the authors' minds.

So, chairs exists and numbers subsist? Is that a common understanding? — Art48

As an ex prof I never thought about it, and I don't recall hearing the expression, but I suppose it could be appropriate.

"In a Noetherian ring, suppose that maximal ideals do not exist. — Pierre-Normand

I asked ChatGPT a question about an obscure mathematical topic, specifically how I might proceed to explore a particular aspect. It gave me an accurate answer as well as a simple perspective I had overlooked. Quite impressive.

I am very knowledgeable about the history of a certain activity. By asking about a specific incident several different but equivalent ways I got several different answers, two of which were correct. I suppose there is a rule of thumb about giving the AI as much info as is reasonable to get the most accurate reply.

I've mentioned this before, but it has relevance here. There was a PhD mathematics student who investigated a class of functions for his original research project. After some time he had discovered several fascinating facts about these mathematical entities, enough for his dissertation. But he was then asked for an example of such functions. It turned out the set of functions was the empty set. I have no idea if he eventually got his degree.

Were mathematics more a kind of philosophy, he might have received his degree even if he proved much about nothing.

Could you give some real life examples of such existences in the real world? — Corvus

Please do. Would some aspects of thought or ideas fall into this category? Spacetime?

:up: Sometimes a little poetry succeeds where pages of prose do not.

↪jgill
Yeah probably. Out of curiosity, would you have a better description of mathematics? — A Christian Philosophy

As a former prof I never gave much thought to a definition. But this is OK.

A conversation is between two entities. I don't see how you separate from Chat with what you have presented. But when I clicked on the link I saw how the conversation developed.

mathematics (which is logic applied to numbers). — A Christian Philosophy

Overly simplistic IMO. Although mathematical proofs are applications of logic.

I tried posting in the general philosophy section an interesting conversation I had with chatGPT about nothingness — Daniel

Is that entire commentary the product of ChatGPT or some sort of blending of what the AI said and what you propose?

Perhaps nothingness is not the “natural” state we imagine; instead, it might be the ultimate state of chaos, paradoxically requiring that something arise within it to achieve even the smallest hint of order or stability. — Daniel

Poetic, but a tad nonsensical.

I was once an observer in a room in which my friend having an intense and sustained conversation with an other who to my eyes and ears was not there — unenlightened

Yesterday, upon the stair,
I met a man who wasn't there
He wasn't there again today
I wish, I wish he'd go away...

When I came home last night at three
The man was waiting there for me
But when I looked around the hall
I couldn't see him there at all!
Go away, go away, don't you come back any more!
Go away, go away, and please don't slam the door... (slam!)

Last night I saw upon the stair
A little man who wasn't there
He wasn't there again today
Oh, how I wish he'd go away...

H. Mearns Antigonish

(couldn't resist) :cool:

Certainly. And as history shows it, we have no idea of them at the present. Only the future will define who is seen as a great person of the start of the 21st Century. — ssu

:up:

As a side note, my lucid dreaming self has limited consciousness compared to my waking self. — GrahamJ

Interesting. My experiences back in the 1970s were that my lucid dream self was more aware than my normal self. I felt myself pure will. The first instance was so powerful my first thought in that state was, "There is something else!"

I retired as a math prof 24 years ago, so practices may have changed. No one in the department used the internet as a substitute for a required text, although those texts cost a great deal. Of course, the WWW was fledgling.

I once experimented using Schaum's Outline in my complex variables course, partly to reduce the cost of the course text from $80 to $18, but also to provide an easy source of facts to accompany my lectures. Mixed results. When students missed class they had to consult with a fellow classmate to get notes or visit my office. Others appreciated the easy access to basic facts and examples.

For a professional these days research depends greatly on one's ability to access preprints in one's area. Books are still useful for grad coursework, but beyond that it's keeping in contact with peers and using the internet to try to stay abreast in one's discipline.

Having said that, in my old age I frequently go to my library and pick out a text published 60 years ago to refresh my dwindling memories.

Constructivism (and my position in the OP) does not deny that objects exist independent of perception. The key point is that our knowledge of such objects is mediated by subjective processes—experience and prior knowledge shape how those objects appear to us. This does not negate their independent existence but highlights the active role of the subject in any knowledge of them. — Wayfarer

I agree. My adventures with the Art of Dreaming were across the spectrum from elemental lucidity to a strong reality, an awareness, more pronounced and sharp than anything I have ever witnessed in normal everyday experience. Here is where degrees of reality has a possible measure. I's unfortunate that achieving these states is so happenstance. For me it was like a second nature.

One of Stephen King's novels has a brilliant description of the central character awakening in a field and smelling growing onions and freshly overturned earth a mile away.

Math and philosophy overlap. I asked the AI a question about a math topic that is somewhat unique, the question being how would it suggest I move ahead. The reply was sophisticated and correct and it was something I had given little thought to. This may be an instance of original thinking arising out of Copilot.

Although it is possible a mathematician somewhere on the internet has suggested this, I doubt it since it pertains to a topic of which I am one of the few professionals having an interest. Perhaps the only authority.

I am keeping an open mind. Fascinating.

At that moment, I had a sudden and inexplicable realisation of the foundational nature of the 'I'. Not myself, as a particular individual, but THE self, the 'I AM' for whom the world exists, without which there is no existence. It suddenly became clear to me that this 'I am' is foundational to reality. — Wayfarer

The AofD experiences I had 40 or so years ago were astounding in several ways, the most pronounced being a wakening into a world more vivid and "real" than normal. As I mentioned I felt I was pure will, but the colors and definitions seemed stronger also. One experience was awakening in a bright desert staring at a large extremely colorful cactus. I moved around it and saw into it from several positions, astounded at the sharpness of definition and brilliancy.

A couple of years ago I suddenly awoke into a world where I was a different person, living in an Irish cottage, looking out a window at a countryside. For a brief few moments I had the sensations, memories and feelings of someone else. Another ineffable encounter with an alternate reality. It was quickly over.

I can't explain why these things happened to me so easily. I was a mathematician and an experienced rock climber - which are not so disjoint as you might suspect.

The "I AM" is the pure will behind the shadows of ourselves.

Thank you for your accounts of personal experiences. I will comment on each over the next day or so.

You are the closest to my mystical adventures. After reading the Art of D, which was brought to my attention by a young fellow rock climber who enjoyed a drug or two, I made the attempt to see my hands, as you did, but instantly I was shifted to an alternate reality - or so it seemed. Stunned, I lay there for a minute or so then got out of bed and walked across the bedroom floor to a closed door, feeling the carpet under my feet and stroking a chest of drawers on the way. I had heard of people walking through closed doors so I gave it a try and it was like going through a thin layer of fog. On the other side a stairwell led down to the living room, and as I started down I was pulled back to normal reality. Later I learned that moving down in that state can result in losing it.

Other adventures followed, and in each I was fully conscious but enabled to violate the laws of physics. How would I describe it in a few words? I was pure will.

already on to a different girl man; the teacher is old news. This other one does in fact like the beach she said, so it might work out super chill. Who knows though — Zolenskify

:up:

What would you like to write/read about? — Amity

Personal experiences relating to philosophical/mystical subjects. Not ramblings about dead philosophers. (My apologies to TPF members) :gasp:

I practiced meditative exercises 60 years ago, thanks to Alan Watts and others and the "Zen Boom", along with existentialism. Then later followed Castaneda's Art of Dreaming instructions with amazing results altering my understandings of reality.

If it is about personal experience, what form of philosophical writing do you think would be best? — Amity

Personal narratives and philosophical speculations thereof. Don't write about Zen unless you are or have been a practitioner, e.g.

(In general, don't write about technical subjects, like QM, unless you have some expertise.)

Spirituality: Secular or Religious? — Amity

Well written details of personal experiences rather than philosophical ramblings should have priority.

Back to the OP momentarily, I think a spectrum of reality is a better approach then degrees of reality, the latter implying a sort of ranking.

Eastern mystical practices, like Zen, revolve about experiences of enlightenment rather than philosophical discussions. Although enlightenment may be ineffable, it would help if more responses on the site were about personal experiences than speculations and searching out what the Greek ancients had to say. I realize this goes against the grain of this site, and that also there may be very few instances of personal revelations, but the discussion is getting nowhere with regard to its title. However, if this is OK with the few participants, and digressions are acceptable, I stand down and watch. Just my opinion.

A professor is not a teacher? I assume you mean pre-college instructor. — jgill

Who said anything about a "pre-college instructor"?
Please, clear the air for me if I offended you. — Zolenskify

. . . is it moral to get with a teacher versus an actual professor? — Zolenskify

No offense. Sorry if I mistook a serious OP for a bit of humor.

and the question always to ask 4) What could possibly go wrong. — T Clark

:chin: