May I ask you a question? How does one come to know this material and not have heard of measure theory? — fishfry

Where does measure theory (surely not taught in high school) intersect any of this? I've used it in various integration processes, the most interesting being functional integration. And Feynman constructed his sum of paths integral in more or less that concept.

there is no definition of the "meaning" of a theorem, and many mathematicians (starting from Hilbert, I guess) think that there is no point in trying to identify the "meaning" as something different from a list of symbols. — Mephist

This must be a significant difference between what you do and what a research mathematician does. Recently I've proven theorems related to compositions of functions in the complex plane, and with each I have a deep feeling, a strong sense of meaning, about the result and how the result comes about. A lot of geometrical mental imagery coupled with the essence to which the symbols point - much like reading literature and realizing all those symbols describe something that stirs the imagination.

However, my theorems are not profound - strictly what Wikipedia calls "Low" interest! :cool:

Probably you think that I completely missed the "meaning" of what a mathematical proof is — Mephist

What you have said about proving mathematical theorems may be the way computers see the process, but most theorems are not proven by computer programs. I don't see the connection to anything I have encountered in theorem-proving. But I am retired and way behind the times in really abstract math. Maybe the world has changed in my absence. Maybe not. I appreciate your efforts to explain, however.

"I agree that that's not all."

Which has to take a prize as an understatement. Is most of your experience in computer science?

And then topos theory is abstract sheaf theory — fishfry

Thanks for saving me the effort of looking it up. That one sentence is enough for me. :brow:

but we do multiply by the product xy which is (-infinity)(+infinity). Is this where the problem occurs? — TheMadFool

Maybe so. :roll:

Yes, of course practically all "normal" proofs are short and all the computing power needed is a pen and a peace of paper. But in reality all computations can be considered to be proofs, right? You reduce an expression in a normal form following some rules (if it's a multiplication between integers the "proof" can be made automatically with a calculator). — Mephist

It depends upon what you mean by "short." Or "normal." In the area I'm most familiar with theorem proofs vary from a few lines to a number of pages in length. And a short proof may be of an extension of a theorem which required many pages of reasoned articulation. I would rather use a pencil than a pen, however, so I can erase my errors or scribbling along non-productive paths of thought!

When you say computations can be considered proofs I'm not sure where you are going. Proofs of what? And "reduce an expression in a normal form" - what's that? Mathematical proofs are rigorously reasoned arguments in logic in which concepts and relationships play significant roles.

There are occasional exceptions in which computers are essential, like the Four Color Theorem in combinatorics. And then mathematicians attempting to verify the overall presentation of proof and conclusion are stuck with verifying computer algorithms and assuming the computers running them do not produce computational errors.

"The correspondence between topology and logic instead, that's one of the most popular and ideas of today's mathematics!"

I'll have to check this out. I've been out in the pasture too long I guess. :worry:

Perhaps there is a little confusion about the symbols ${{0}^{+}},{{0}^{-}}$

They usually refer to approaching zero from the right (through positive numbers) or the left (through negative numbers).

. . . but mathematics need computations for proofs. — Mephist

I have conjectured and proven lots of theorems - some quite challenging - that do not require computations beyond basic inequalities and a little arithmetic of complex numbers. However, creating examples and imagery in the complex plane usually requires programming skills and a computer. So, even if not for proofs, computations are necessary in many areas of mathematics. :cool:

Just when I think I've seen the most :scream: math ideas presented on this forum, I am pleasantly surprised :smile:

(fishfry: how did you post that graph? Are you a forum subscriber?)

. . . all physical experiments are calculated with integrals over open sets. — Mephist

Huh? :roll: Really?? all?

One way to define "constructive physics" is simply to say, "it uses constructive mathematics". But definitions of the latter sometimes arise principally from avoiding the LEM. Another tack is to avoid non-computable numbers. Or simply to state that experiments must be conclusive in a reasonable finite amount of time. I'm not sure what you two are referring to here. But I haven't read all the thread.

This is a big, big controversy in current physics — Wayfarer

I'm aware of that. But I wonder if string theory might be considered a metaphysical subject amongst philosophers who are not necessarily scientists/mathematicians? And if so, if experimental evidence did arise, would this, technically, be considered metaphysical actuality?

Forty years ago I decided to attempt Castaneda's Art of Dreaming. Amazingly, his instructions (Don Juan) worked the first time and I "awoke" in what seemed to be the normal world, but with colors, sounds, etc. accentuated. I would enter this state of mind usually during a hypnagogic interlude. No drugs were involved. Today, these are all pleasant memories.

This qualifies as the quantified theory of love? — god must be atheist

Ha :smile: Just curious if a physicist might jump in here. I don't think there are simple answers.

But I wonder about string theory or similar conjectures, and whether they are considered metaphysical whatevers? If so, if one of these strange theories were verified by experimentation, would they become metaphysical actualities?

The metaphysics in these topics is all over the place - and that's suitable for this forum. Is String Theory metaphysics? Or any other far-out "theory" which evades verification? Perhaps String Theory could be called metaphysical actuality?

Personally, I would be relatively happy if someone describes the integrative measures that are employed in Feynman's path integral. I'm familiar with functional integrals, which involve measures on sigma algebras of "points" (functions) in a space, but the "DX" in the Feynman "sum over all paths" integral is still a kind of puzzle to me.

:chin:

Was it Cantor who said the rational numbers are like the stars in the night sky and the irrationals are like the darkness in the background? Perhaps this has been posted before.

Falsified theories are replaced by theories of greater explanatory power. — Banno

The replacement of a theory due to counterexample (original theory wrong), vs the replacement of a theory due to the development of a better, more encompassing theory (original theory correct, but superseded). Business as usual in science.

You make the distinction between theory and statement.

I'm a mathematician who studied a decent amount of logic — fdrake

I'm one who hasn't. Complex analysis here. And you? :cool:

I am talking about 'Reason' - she is the person, the god, whose prescriptions our faculty of reason. . . — Bartricks

OK. Now that I see the thread is theological I understand. Carry on! :cool:

Reason determines what's true — Bartricks

At one time it was reasoned that a bolt of lightning was due to an action by Zeus. It might appear then that "reason" is not necessarily the test of truth. :gasp:

Then why did you mention Zen? If neither you nor I know anything about it, why mention it as if it had some importance? — Bartricks

I never said I knew nothing about the practice. In fact, I do. I do not practice it now, however. As for importance, serious devotees reach a mental state in which empty awareness or no-thingness becomes "real" and may seem, to a non-devotee, to conflict with pure reason. Something arising from emptiness is not necessarily nonsensical.

Is it possible there are aspects of reality that may be beyond what we consider reason?

Or keep electron and positron at the same point in space and you will walk through them, would not be able to see them, detect or interact with them in any way. Effectively, practically, they will be nothing, — Zelebg

Like Zelig? :smirk:

We no longer follow visual procedures in mathematics. — alcontali

Not true. For example, I just posted a research note in which I gave what is primarily a geometric (visual) argument that the iteration of a linear fractional transformation form converges to a limit for a portion of the complex plane.

The second point is logical. That a proposition is falsifiable is not the same as it's being true; and hence, there will be verifiably falsifiable propositions that are false, yet unfalsified. — Banno

You might want to elaborate with an example. In appearance, it looks like word salad. I would think that if a proposition is falsifiable it is not the same as it being false. In other words there is a procedure for determining falseness, but it hasn't been applied yet. I haven't been following the thread, however, and must be missing a technical definition. Confusing. :worry:

Doesn't your reason - your faculty of reason - tell you that nothing comes from nothing? — Bartricks

Is reason axiomatic? How do you think it develops or arises in the human mind? Can it change as a culture changes?

I don't practice Zen. Look it up on Wikipedia.

And again: can something come out of nothing? — Bartricks

I would guess not, but I am not certain. Zen has quite a bit to say about no-thingness.

It's a self-evident truth of reason that every event has a cause — Bartricks

Hence, a thought you hold true must always be reflected in nature. Metaphysical actuality.

But what about the billion or more people who live at, or slightly above sea level, will they come and join you, when you move uphill? — Punshhh

I live at one mile elevation on the prairie. Lots of room up here. But the surfing sucks.

So, every event must have a cause. — Bartricks

Why are you certain of this? Because that's the way the world works now?

This means that taking a square root is not a valid operation — Metaphysician Undercover

Define "valid operation." You should have been around to make your current argument about 1700BC when the Sumerians were calculating the square root of two (and its reciprocal) on cuneiform tablets. They would have appreciated your perspective. :smirk:

I don't understand your definition re: metaphysics. What are things? We seem to be in a downward spiral here.

On the other hand, for a little clarity:

Des Bosses to Leibniz (1700s): "Monads are metaphysical actualities."

Now, that makes sense. :cool:

Speak to metaphysics, please. Define "actual" in that context. — jgill

Metaphysics is a branch of philosophy, within which "actual" has a technical meaning that distinguishes it from "possible" and "necessary." — aletheist

If I say that I have an apple, what I usually mean is that I have an actual apple. If I posit a set of apples in the strictly mathematical sense, then I am talking about something that is logically possible, but not (necessarily) actual — aletheist

"Metaphysics is a branch of philosophy, within which "actual" has a technical meaning that distinguishes it"

Speak to metaphysics, please. Define "actual" in that context. :nerd:

x=x

Leibniz: "It is what it is."

Clinton: "It depends on your definition of 'is'."

:chin:

That is not what I mean by "metaphysical actuality." I just mean the modal property of being actual, rather than merely possible or strictly necessary — aletheist

So, combining "metaphysical" with "actual" means someone is thinking a metaphysical thought? Or does the expression imply an interaction with physical reality? I am going on a classical definition of the expression. What do you really mean? Please clarify with examples.

I am not a philosopher.

Thanks. :chin:

It is not; as I said, mathematical existence--including the potential infinity of the natural numbers--is not metaphysical actuality, it is logical possibility — aletheist

Metaphysical Actuality: The philosophical position that thought becomes actual by becoming concrete. Subjectivity, the "I" has constitutive validity, having sole omnipotence.

This seems to me an extreme position. How then does it interact with historical actuality? :chin:

Interesting links. Thanks.

I've dabbled in mathematical vector fields for years, particularly time-dependent fields. If it were possible I would post some intriguing images, but there are problems doing so. We'll see. :nerd:

I do not understand you or why you are weeping with laughter — Bartricks

I got carried away by your comment that physics doesn't investigate reality, which is the domain of philosophy. I cited the "x=x" thread because, to a novice philosopher, the law of identity (Leibniz's "It is what it is") seems beyond refutation and axiomatic. How can it contribute to a further understanding of reality?

I don't agree with your comment. Physics most definitely investigates reality. The fact that scientists may avoid discussions of causality simply means that by doing so they can achieve a better understanding of physical reality.

But I reacted excessively. Sorry.

About the moon which is not there breaking into pieces which aren't there either? :nerd:

An extreme version of reincarnation? Or is it?