It will have a title. — gikehef947

I thought it already had one? The mystery deepens.

If a bitcoin can go from $40 to $50,000 in a couple of years anything is possible.

I’m in love with my own shadow, and I don’t want to be. I want a divorce. I want her to leave me alone. She follows me where ever I go. She intrudes. People say I’m entitled to my own shadow. So what? What am I going to do with her? — James Riley

Get out of the sun of introspection.

The geometry used [Non-Euclidean] is the one developed to suit the application, it's produced for a purpose. With the conflation of time and space, into the concept of an active changing space-time, Euclidean geometry which give principles for a static unchanging space, is inadequate. Hence the need for non-Euclidean geometry in modern physics — Metaphysician Undercover

Wow! I am really impressed to realize Omar Khayyam (1048-1131) had the perspicacity to realize his efforts at Non-Euclidean geometry involved notions of space-time. Thanks, MU. I would not have guessed. :chin:

Curious. Why remove his thread on academia?

But many (maybe the preponderance of) mathematicians regard the axioms for particular fields of study not to be merely arbitrary, but rather as meaningful and true. — TonesInDeepFreeze

True. The degree of meaningfulness in direct proportion to the number of hours spent working in that discipline. In real and complex analysis one takes (as non-arbitrary?) and axiomatic the definitions of convergence , continuity, etc. given by several mathematicians, including Cauchy and Weierstrass (my math genealogy ancestor - along with 36K other descendants). :cool:

I think that might have been sexist. — frank

Oh oh. I apologize to Ms Xtrix. :worry:

The issue, at least as some of us see it, is the degree of suffering experienced by us and other humans - including that which is to be experienced by future generations - while alive — javra

Finally, an element of intelligence in this conversation, rather than hysteria. Then one gets into the nitty gritty of degrees of suffering and how to possibly mitigate some of that.

This isn't a problem we can be optimistic or pessimistic about -- I don't think that applies. We're either going to make it happen or we're dead — Xtrix

But again, no scientists are saying "we're doomed," and neither am I — Xtrix

How many of you actually live any philosophy? For me, I've followed am existential path, hoping to create meaning in my life. At times I have succeeded. The precise use of language is important as T Clark has indicated.

Having left the profession before remote learning became fashionable, I am impressed with this thread. An ideal arrangement having two highly competent instructors and one focused student. :cool:

One might expect Metaphysician Undercover (Unknown) to chime in with "inherent order", but he might be out of his depth.

Well done! Handy reference for an oldtimer, Too. :cool:

If there is order which inheres within a thing, then that order puts a necessary constraint on future possibilities of order, due to the continuity (causal relation) which we assume to exist between past and future. — Metaphysician Undercover

Interesting ideas, but a tad too ethereal for me. I prefer the solidity of pure mathematics. :cool:

Mitigation and adaptation. The former requires worldwide commitments. The latter can be dealt with by individual nations. Which do you think has the better chance of succeeding?

The simple instructions he gave worked on my first attempt. No drugs. It was an astounding experience and I thought "there are other realities." As time went on I found there were different degrees of the experience, from a very close approximation of normal reality to bizarre escapades far removed. I recall walking across the rug in my bedroom, feeling the fibers underfoot, and tapping a chest of drawers with my fingers. Then walking slowly through a closed wooden door, as if a thick momentary fog.

On that first experience I also thought, "this is how religion began."

It seemed one becomes an embodiment of pure will or intent.

Yes. Many years ago. Castaneda's Art of Dreaming. Extreme clarity of images, in full control - pure will it seemed.

So, as much as I see the claim that "pure mathematicians" are motivated by aesthetic beauty, as opposed to real world concerns, I don't see that any such concepts as being produced purely for aesthetic beauty actually exist in mathematics. — Metaphysician Undercover

I can attest this (my) mathematical concept was produced purely for aesthetic beauty. The fact you don't see a string of symbols is immaterial. It is found on a mathematics page in Wikipedia.

The question is whether those principles ought to be derived from pure imagination, or ontology — Metaphysician Undercover

Is there such a thing as "pure imagination" that does not arise ultimately from observations and experiences in the physical world?

Well there's really no alternative I can see -- so you're either wrong or we're dead — Xtrix

Oh my. I can be wrong and we all will still die. But not all from climate change.

That people like you and I and others aren't pushing hard enough for it. — Xtrix

I could push every minute of every day, along with everyone else, but that's probably not enough to keep those gulf streams warming Europe in place. London and Dublin are further north than Calgary, Canada, and are roughly the same latitude as southern Alaska. Get your woolies out, mates!

For mathematicians, the words abstract and pure are frequently taken to mean the same thing. Here are comments from the Wiki page:

Pure [abstract] mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles. . . .


. . . It follows that, presently, the distinction between pure and applied mathematics is more a philosophical point of view or a mathematician's preference than a rigid subdivision of mathematics. In particular, it is not uncommon that some members of a department of applied mathematics describe themselves as pure mathematicians. — Wikipedia

As a working math person I rarely thought about these philosophical issues, But "formal abstractions" in foundations appears to be a very sophisticated game. Whether this game is entirely separated from the material world might depend on how one interprets "separate".

And then there are those occasional "aha!" moments when a physicist uses a mathematician's abstractions to describe a physical process well enough that accurate predictions ensue. Accidental, or subconsciously generated to match physical reality?

Is it already too late? — Xtrix

I suspect it might be if one thinks of significantly slowing the process. I don't see the nations of Earth coming together in a meaningful way, but I could be wrong. Even were they to do so it might be best to prepare for change wherever possible.

I was a meteorologist many, many years ago. Too long ago to be of any use.

Just saw this, apropos: — Xtrix

"The earth may be parched and sweltering, but all is not lost, for Americans finally got in the game."

What a joke.

Explicitly constructive mathematics goes back at least a hundred years, and with roots in the 19th century too. It has great importance toward understanding foundations. I think interest in it goes well beyond any need for assigning research topics. — TonesInDeepFreeze

My comment about abstraction wasn't really in reference to foundations, just a general reflection on the profession. I mean no disrespect to constructive mathematics. I recall my advisor fifty years ago talking about classical analysis being somewhat mined out and mathematicians moving toward generalities and abstraction to find new territory. He thought interest in the classical might come back at some point.

(Actually, I'm a fan of Brouwer, more so of Banach, whose fixed point theorem I extended to suit my special interests.)

jgill This place is corrupting you! — fishfry

Makes my head spin. Thanks for opening my eyes a bit to formalized set theory and for your patience!

In published mathematics research papers estimates of serious mistakes run from ten to thirty percent.

(of course, my papers were error-free. :cool: )

(I've read somewhere that the percentage might possibly be highest in foundations. I'll see if I can find the comment.)

This stuff just gets more bewildering as time goes on. :worry:

(One reason math has become so abstract is that classical areas of investigation have been "mined out". Professors need suggestions for research topics for their PhD students. So, create new definitions and/or generalize.)

What about sending a signal into the past or future rather than a macroscopic object? — Enrique

This is a really good movie incorporating your idea: Prince of Darkness

The connection is the fact that the axiom of choice is equivalent to the law of excluded middle, — sime

"And of course, we know that LEM does not imply AC, since we know that ZF is consistent with ¬AC while LEM holds." (MathStackExchange) :chin:

Excellent commentary. Sitting in the bleachers awaiting MU's reply.

On the other hand, symmetry = invariance under transformations.

for a set is normally specified as a collection of things which satisfy a given predicate . . . — sime

. . . nobody knows what a set is — fishfry

Don't feel badly. As a professor of math I rarely thought of sets that did not satisfy this definition. In passing I thought of Russell's paradox as quirky, requiring set theory be a little more sophisticated. Living in a state of "high school" naivety all those years has been devastating. How can I recover, now I am near the end of my days? :cry:

Mathematically, it might well be the case that the number of grains of sand in a heap is neither finite nor actually infinite, but indefinitely large — sime

As a mathematician who has dabbled for years in complex analysis - a branch of math that involves the limit process, calculus, etc. in the complex plane - I have rarely if ever interpreted infinity as a kind of "number" but as shorthand for "unbounded", which correlates to you always finding another grain of sand.

As for inherent order, I admit that writing {c,a,b}, although permissible, just doesn't seem right. I would always prefer {a,b,c}. :cool:

You know it's an oxymoron to talk of a symbol without meaning — Metaphysician Undercover

True enough. I should have said random configuration of line segments or something like that.

It's an intentional act, so there must be reasons, therefore order. — Metaphysician Undercover

Are all acts founded on reason?

Is there an axiom in set theory that requires the display of elements of sets to be done in inherent order?

Therefore there is an order which inheres within that set of abstract objects, necessitated by the meaning of the objects — Metaphysician Undercover

But as simple symbols, rather than meaningful symbols, they may have no IO or a different IO. If I make up three random symbols from finite lines, say, would you then state the order in which I created them gives IO to the set?

It is evident, that at the most fundamental level, living beings make use of inherent order, by creating extremely complex molecules, etc.. — Metaphysician Undercover

I brought this up earlier, the notion that "ordering" and "order" were not the same, saying that order had to do with biological and other systems, rather than listing. How order relates to ordering the elements of a set is unclear.

It's unfortunate the word "block" is used in this regard. It immediately configures the mind in a certain way that might be hard to ignore when contemplating time.

I like your new photo. Are you wearing your hair in a different fashion?

You are ignoring the fact that I repeatedly said that we see the inherent order without apprehending it with the mind — Metaphysician Undercover

You are assuming the existence of an inherent order that lies beyond conscious recognition. Is there another aspect of mind that might register this phenomena? Is the fact we can discuss IO due to this possibility?

I suspect the image of a space/time "block" is just as misleading as the image of the Earth sinking into a kind of mesh representing how mass deforms space. Right along side of the notion of our consciousness "collapsing" a wave function. All are attempts to make palatable mysteries of reality that may be most accurately representable by mathematical structures.

"What are the philosophies of mathematics that underlie the movements in math education based on math trails/walks?" — Paul Fishwick

I went to lunch today with retired colleagues, one of whom was a professor of math education. I asked them about math trails or walks and neither was familiar with the notion. So even academics can draw a blank on the subject.

If there was a philosophical area behind the concept I think it might have shown up here by now.

There may be no single philosophy of mathematics that is situated empirically in seeing math in everything. — Paul Fishwick

It may be a matter of perspective or personality rather than philosophy. In my years as a math prof I don't recall any colleague particularly interested in seeing math in play around them to any prolonged extent. But there are probably some out there who are like that. However, I'll bet Max Tegmark is not of that ilk!

What you refer to might be ontological, but I doubt it is an aspect of philosophy of mathematics, which
seems to focus on foundations, math systems, logic, Platonic vs non-Platonic, etc. But I could be entirely off base here, and welcome illuminating comments. :cool:

So, putting all of the pedagogy and math trails aside, what exists within philosophical discourse that promotes this way of seeing? — Paul Fishwick

Beats me. I was a math prof for years and never had an interest in seeing math in everything. :chin:

Mathematical Universe

This might be of interest. I'm doubtful, myself.