Math PhDs 2017-18 in USA: 1,960 ... 29% women.

Something about the name, "Metaphysics Research Lab," strikes me at the same time as hilarious and terrifying. — tim wood

we can avail ourselves of various treatments for vagueness — Caldwell

There are vaccines being developed in the MRL to protect against this very affliction. :cool:

Will be = must ? Too deep for me.

Ambiguity is part of philosophy. That's what seperates it from science — Don Wade

Praise the Lord for that distinction. :wink:

Maybe this is somehow connected to your question. Maybe not.

and quantum physics is the frayed edge of reality, where existence bleeds into nothingness. — counterpunch

But there is some lovely mathematics that relates to this arena of exsanguination. Particle spin associated with the delightful correspondence between points in R3, quaternions, and the matrices of SU(2). I'm dabbling in that now. :cool:

Here is a question and answer from Quora Digest that was in today's link for me. Not a lot is said about the amount of time involved, but it may be shorter than the end of the universe:

David Joyce
July 5, 2020
Ph.D. in Mathematics, University of Pennsylvania (Graduated 1979)

Could there be a computer program, that would take any mathematical expression as input (for example, Fermat's Last Theorem) and then compute a formal mathematical proof as output? How difficult would it be to write such a program?

"There is an algorithm that finds in a finite amount of time a proof of every statement that has a proof, and runs forever for statements that don’t have proofs. The algorithm is just do a breadth-first search.

You can improve on that by searching for a proof while searching for a proof of the negation of the statement. If there is such a proof or a counter-proof, then it will be found. But if the axioms for the theory don’t support either a proof or a counter-proof of a particular statement, then the algorithm runs forever on that statement.

If you have a complete theory, such as elementary Euclidean geometry, then for every statement, either the statement has a proof or its negation has a proof. But if the theory is not complete, such as group theory, then there are statements which don’t have proofs, and their negations don’t have proofs. Number theory is also an incomplete theory as Gödel showed in his incompleteness theorem"

Complete Theories

:up:

It seems simple to me. Just consider that time and place in history. Stand fast for Christ and His teachings. But these days, how much of a Christian are you? There are degrees of commitment.

I suspect you are a physicist or do you write software? — turkeyMan

Click on my icon. Your "summary of all mathematics" caught my attention! :roll:

There is another software project that i actually completed (unlike the bartering app) and it is a 10 dimensional linked list relaitionship finder. It really needs alot more work but it is summary of all mathematics. — turkeyMan

Great! You must be a fellow mathematician. :cool:

On this thread, the posts imply nothing is as it seems. :cool:

. . . but don't start with primary sources. — Kenosha Kid

That's what my philosophy prof told me in a senior-level class when I went to his office and complained that I was having real problems trying to understand the writings of one of the classical European philosophers I was assigned to report on.

I'm at my wit's end in trying to publish something in an academic journal. — god must be atheist

Publishing in such journals may be difficult if the author is not in academia. This has been discussed elsewhere in the forum. Good luck.

“The Moving Finger writes; and, having writ,
Moves on: nor all thy Piety nor Wit
Shall lure it back to cancel half a Line,
Nor all thy Tears wash out a Word of it.”

Omar Khayyam (ca 1100AD) - As a mathematician he worked on continued fractions, a subject I studied fifty years ago. :cool:

. . . but philosophy journals, as distinct from popular magazines like Philosophy Now, are mainly read in philosophy departments. I would have thought that getting something published in them, if you have no history of publication or academic affiliation, would be quite a challenge. — Wayfarer

Yes, I'm curious about this. In mathematics, academic institutions have a very powerful influence of what gets published in important journals. The reviewers are mostly faculty members or work in jobs that are roughly equivalent.

Wow. Is Denmark that bad? :scream:

Have you published before in a high impact journal? Good luck.

For me, this is a very interesting thread. It's more relevant to my own experiences as a math prof than other threads about mathematics, especially those that argue about the equal symbol and and the Platonic characters. I am curious about heuristics as a research methodology. The research I have done involves speculations about conjectures and imagery that crops up in my computer programs, plus analyzing examples - all very loosely coordinated - until an idea pops into my head. If there's a methodology there it is loathe to reveal itself.

But, then, I was an average math person and not in the upper echelons where things may go differently. To give an example, here is a recent very informal note I posted on researchgate that explains to some extent an intense iteration process in the complex plane. It's of no consequence. I do these things like other old geezers carve wooden ducks, keeps me occupied since retirement twenty years ago. But it does show imagery that sparked the actual math: It is also full of the kind of heuristics I assume might give Lakatos the Vapors.

Tunnels to Centroids of Attractors

We cannot have any awareness of time without memory, — Garth

There was a movie some time back in which the central character had no short-term memory, but solved some sort of mystery, taking copious notes continuously. Anyone recall?

Well, philosophy is certainly good for nothing, at least on TPF:

Nothing! A Conceptual Paradox!

:cool:

Well, that's certainly a serious undertaking. Much hinges upon your findings.

What that does do, however, is make mathematical demonstrations heuristic in Lakatos' sense; more about displaying the concepts to a sufficient degree of obviousness than mandating that all proofs have every step of reasoning spelled out symbolically. — fdrake

Heuristics in mathematical expositions is relative to an intended audience.

Yes, an expectation of the reader filling in the gaps is common. It would be dreadful if that practice were abandoned! :cool:

In mathematics, at an opposite pole are extremely formal proofs by computer algorithms — jgill

Yes indeed, that's also why Lakatos' book is really interesting! He argues against formal mathematics byt his method of "proofs and reputations" may very well be applied to formal mathematics as well.

I am really interested in the use of heuristics as a research methodology :blush: — Twinkle221

I was speaking of completely automated proofs by computer programs. Formal mathematics is simply that employed by research math people these days. You know all this of course. I've now read a bit about the topic here and I suppose what I have used might be called naive heuristics, in light of all the various types of heuristics described in Wiki.

For me, heuristics in math are not problem-solving rules, although at elementary levels that definition would apply. From my perspective, in research and presentation the word means simplifying by avoiding rigorous logic at various steps, staying holistic to a conclusion. In other words, hand-waving. I guess I never considered it a research methodology other than speculation involving examples.

Is that you in the physics lab explaining quantum entanglement?

You have my attention.

Without trying to answer for jgill, I don’t think looking for ‘the wise man’ to solve our lack-of-knowledge issues is realistic. Wisdom is demonstrated in collaborative achievement - in the imaginative, understanding and non-judgemental relation between insufficient perspectives - and recognizing that no man alone can embody this faculty is as important as seeking it out. — Possibility

Well, yes, that's what I meant! Admirable work, Possibility. :grin:

jgill Are you suggesting, Gilly, that the wise man doesn’t exist, or that The Mad One’s description of him is false? How would you describe the wise man? — Todd Martin

A fictional character. :roll:

Nothing is a very clear concept. Is the lack of something. — Helder Afonso


A lack of anything. Everything lacks something. My dog lacks a tail. — Kenosha Kid

Well, you would have to nit-pick! :cool:

Markus Gabriel: the Wise Philosopher

Wikipedia: "In an April 2020 interview he [Markus Gabriel] called European measures against COVID 19 unjustified and a step towards cyber dictatorship, saying the use of health apps was a Chinese or North Korean strategy. He said the coronavirus crisis called into question the idea that only scientific and technical progress could lead to human and moral progress. He said there was a paradox of virocracy, to save lives one replaced democracy by virocracy."

Who would pay money for this guy's lectures?

A wise person isn't confined to specific disciplines but has a fair if not complete grasp of all that can be known and the hope is that with such a broad understanding of the world, fae will provide the best possible answer/solution to the questions/problems that the world has to deal with. — TheMadFool

A fictional character.

I never knew heuristics was so complicated. I frequently used what mathematicians consider heuristics when teaching, particularly in a course in complex variables. For instance, the maximum modulus principle is easily explained on the blackboard by drawing a simple picture, and this heuristic motivates an actual proof. And when the question of a physical interpretation of the derivative of a complex function arises, another picture on the blackboard demonstrates its meaning as a contractive factor.

As for published papers, heuristics can be quite motivational for tempting other professionals to read deeper into your work where things get more formal.

In mathematics, at an opposite pole are extremely formal proofs by computer algorithms, when that is possible, with nothing presented except printed symbols. Anathema for most mathematicians at the present time, but possibly what lies ahead.

Of course they are identical — SolarWind

Your original premise seems to resemble the ongoing arguments on this forum about identity and equality. Does A=B mean that A and B are the same? Is 2+2 the same as 4? That's as far as I can get in your proof. But it's original thinking.

Ya gotta be kidding. That's a rational argument? :roll:

Since the worlds are both altogether different and materially identical — SolarWind

If these are parallel worlds that are identical, then they have the same "histories", and person A in one world would have the same memories and history as person A in the other world. Your statement above doesn't make sense.

But from a brief perusal on the Internet, I see that that usage is found mainly in high school level algebra texts (and "college" level that is seen in the examples to be really review of high school level). — GrandMinnow

Of course. In my years as a prof the only times I recall discussing the equal symbol is in elementary set theory. If we had had an actual course in foundations or even set theory It's not likely I would have taught the course. I looked in an advanced text, Introduction to Topology and Modern Analysis by Simmons, which I consider an exemplary work of clarity, and he only gives the definition of "=" at the beginning for sets.

It is often wise to be wary of high school level explanations and terminology that need to be made rigorous and even corrected by rigorous mathematical treatments (for a salient example, the definition of 'function'). — GrandMinnow

Thanks for the advice. A bit late, however. :cool:

Replying to MU:

In common, pervasive usage in mathematics, as I mentioned, a formula

T = S

is true (or satisfied) if and only if 'T' and 'S' refer to the same object.

The reason you are not familiar with that fact is that you are not familiar with rigorous mathematics and especially as mathematics is treated in mathematical logic. — GrandMinnow

Gosh, you are so confident!

Wikipedia on Equation: "There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variable. A conditional equation is only true for particular values of the variables."

This is how most practicing mathematicians (possibly not those engaged in foundations) see things. Show us your background in mathematics, please.

I have no problem with philosophers talking about the big issues. The question is, should anyone pay attention to what they say?

Is there anyone knowledgeable who would be willing to summarize, in layman’s terms, the communications breakdown in this argument? — TunnelVision

I bow to the patience and perseverance of those who have replied.

What I'm driving at is that just as a car or an elephant is said to exist in the universe, a unicorn or god too exists in the universe. True that one exists in the physical and the other in the mental but both worlds are, at the end of the day, part of the universe — TheMadFool

I agree. But some would separate the physical universe from its mental counterpart and suggest "universe" is only the physical universe. Same with "objects": physical and mental. In the quantum world things become fuzzier it seems.

<Something> + <Opposite> = <Nothing> — Harry Hindu

I think you've broken new ground here. :roll:

(3) It was claimed that '=' has two different senses, for example:

2=2

vs,

2x=x+3

But those aren't different senses of '=' — GrandMinnow

These represent two types of equations in mathematics. But you are correct in a fundamental sense of the symbol "=". (just another small reason I've stayed away from phil of math - angels dancing you know where)