I would not want to be in the shoes of Republicans who will have to answer to their moderate base — 3017amen

Many of whom feel the president did something wrong, just not cause for removal from office.

A conclusion cannot be incompatible with the premise — Metaphysician Undercover

And sometimes a conclusion is indistinguishable from the premise.

:roll:

I shudder to think of what would happen here if the posters on this and other threads with minimal mathematical knowledge apart from set theory and logic were to launch investigations into subjects like functional integration or even metric spaces or advanced calculus. But maybe there is a hidden reservoir of mathematical understanding just waiting for opportunities for expression. I may try starting a thread and see what happens. I know several of you have significant mathematical depth. But others? Not so sure. :smile:

Did you read what I wrote? Energy is a property of the "field", transmitted through the field. The field exerts a force on the particles. You do not see that the "field" is therefore a "substance"? Also, the field exists between the object which creates it, and the particles effected by it. Do you not see that the field is therefore a substance. — Metaphysician Undercover

Earlier MU: "Ether is necessary to account for the reality of waves. A wave is in a substance. We can deny the reality of these waves, but then fields and wavefunctions don't represent anything real. Observation attests to the reality of these waves. If the waves are real, then so is the medium in which they exist."

Substance: the real physical matter of which a person or thing consists and which has a tangible, solid presence

Matter: physical substance in general, as distinct from mind and spirit; (in physics) that which occupies space and possesses rest mass, especially as distinct from energy.

---------------------------------------------

Argument here is hopeless. Is there a real, live physicist who will enter the discussion and untangle this mess? :roll:

Much of everything we know about mathematics was developed by that school. — Michael Lee

A bit of overstatement. But entertaining post. :cool:

It appears like you would not believe what I produced anyway. — Metaphysician Undercover

No. But I would believe what Feynman produced. All you've been saying is you believe there is a physical substance through which waves travel, even electromagnetic impulses. I think the "medium" to which you refer is a metaphysical medium.

I don't understand "backward" causation but I feel the future cannot cause the past. — Andrew4Handel

Unlike some on this forum I don't have many weird ideas. But I've had the sneaking suspicion that in some way or shape the present might influence the past. Many-worlds notions might come into play. Just speculation. :nerd:

Hence, mathematical analysis could suffer from the same fundamental interpretation problem surrounding infinity — alcontali

Classical real or complex analysis: very doubtful. Soft analysis: no telling where that is going. :cool:

Thus, we are justified in believing very firmly that 2 + 2 = 4. — Bartricks

Wrong. 2+2=1
Figure it out.

You just don't learn, do you? :roll:

Please do. I am curious. :chin: — jgill

Without taking the time to research specifics, I can tell you the simple idea — Metaphysician Undercover

So you really can't back up your statement. OK :roll:

I believe that some physicists such as Feynman have produced very convincing arguments which demonstrate that electromagnetic fields must have real physical existence, i.e. substance. If you're not familiar with this, I could look it up for you — Metaphysician Undercover

Please do. I am curious. :chin:

However, physical existence doesn't necessarily mean a substance as medium. It just means it exists and interacts with the physical universe. But I could be wrong. Probability waves are a lot more abstract.

The premises of life: an ever ambiguous concept — The Abyss

It took 27 posts to reach an essential point of clarity. My biggest argument with philosophical discussions is lack of clear definitions. What are some or many of these "premises?" List some, and then others can reply intelligently (maybe).

At the age of 83, the best advice I could give is

Unleash your inner existentialist — The Abyss

Mathematicians who work in mathematical analysis - particularly what is called classical real or complex analysis - don't often say such-and-such is infinite, rather they say a process tends to infinity, provided, for every bound one might conjecture, the process eventually moves beyond that bound. For example, suppose we state

$\underset{x\to a}{\mathop{Lim}}\,F(x)=\infty$

We may say, "F(x) goes to infinity as x goes to a", but what we mean is that for each
$M\gt0$ there exists a positive number $\varepsilon =\varepsilon (M)$
such that

$\left| x-a \right|\lt\varepsilon \Rightarrow F(x)\gt M$


$-\infty$ can be used in a similar manner. And there are comparable definitions for complex valued functions in the complex plane.

It doesn't go much beyond this sort of (Cauchy-Weierstrass) definition in analysis. Orders of infinity and the like normally don't appear in the literature. However, soft or modern analysis does move in the general direction of set theory and algebra. And set theory is a different story; most of the posts in these kinds of threads pertain to that subject. :cool:

2+1=3 is contingently true; not necessarily true. But don't fret. The entire thread is unnecessary babble.

:roll:

↪Pfhorrest It is still producing no mirth in me. — Bartricks

To be mirthless at such wit means you are behind the times. :smirk:

Physics tells us that even bricks are nothing more than probability waves smeared across the universe — fishfry

Aha! So, finally, a resolution to the question I posed about PWs on another thread. The medium through which they travel are brick roads. And they culminate on the shores of the Emerald City!

Thanks! :nerd:

Analytic philosophy is just industrialized thinking. Robots will do it better in a few years. — Pneumenon

Perhaps the same will be said of mathematics. There's a crew working indirectly on that project on this very forum. :smile:

I still like the idea of a metaphysical medium through which probability waves travel.

But, assuming the PW is itself a metaphysical entity, then it is a metaphysical actuality since it "describes" a physical observation. A previous thread mentioned this notion, and I find it entertaining to contemplate. :cool:

2+1=1 mod(2)

Thus your entire argument falls apart. Sorry :cry:

Yes, there is a wave function, and yes, the function is probabilistic, but that doesn't make the function a 'wave of probability' traveling through a medium — noAxioms

"By analogy with waves such as those of sound, a wave function, designated by the Greek letter psi, Ψ, may be thought of as an expression for the amplitude of the particle wave (or de Broglie wave), although for such waves amplitude has no physical significance." Encyl. Brit.

I was suggesting a metaphysical argument. Interested in seeing what it would provoke. :snicker:

QM does not posit waves of probability, through a medium or otherwise. — noAxioms

Of course it does, but not in a physical sense.

From Physics.Org: "At the heart of quantum mechanics lies the wave function, a probability function used by physicists to understand the nanoscale world. Using the wave function, physicists can calculate a system's future behavior, but only with a certain probability. This inherently probabilistic nature of quantum theory differs from the certainty with which scientists can describe the classical world, leading to a nearly century-long debate on how to interpret the wave function: does it representative objective reality or merely the subjective knowledge of an observer?"

What about thinking of tangent vectors to a circle in the complex plane rather than a sphere? But I'm not following closely. :sad:

The Wiki article on Analytic Philosophy is a shocker — Banno

I'm no philosopher, but the article looks good to me. What in particular do you find in error? Be specific, please. :chin:

I still haven't come to a conclusion regarding metaphysics in the realms of science and mathematics. At times I feel that only scientists who are involved in or knowledgeable of research into a particular topic are qualified to delve into the metaphysics of that topic, and at other times I think that this requirement is too stringent.

What do others think? :chin:

What is the medium through which probability waves in QM travel?

How about it, physicist out there? Clarify the idea that MU advances? Waves in fields create particles? Good luck with the metaphysics of fields. :nerd:

One senior level college course, taken before the time of the dinosaurs.

A PhD in math, during the time of the dinosaurs.

Out to pasture, now. :cool:

Empirical evidence indicates that there are waves and this necessitates the conclusion of an "ether" or some such substance which the waves exist in — Metaphysician Undercover

I would like to hear this from a physicist. :roll:

I can summarize. Short answer is that these days you can do logic via category theory; and when you do that, you get intuitionist logic (denial of the law of the excluded middle (LEM) and all that) in a natural way. — fishfry

I really appreciate your explanation. Thank you. I looked briefly at your first link to get an idea of the univalent approach. I was completely unaware of this, being happily non-constructive at times! :smile:

I said, in time Wikipedia will be cited, because I think there will be significant changes to academic vetting. Not now. I never had any problems publishing articles in math journals, but when you get a chance to look under the surface in a discipline you see the problems - and the profit motives. The scholar does the work, the reviewers do the work, and the institution pays the publisher, then other institutions pay for subscriptions. Nice racket. Has that changed? I know all the reasons for the system, but it is mildly corrupt.

On the other hand the proliferation of online journals with weak refereeing standards exacerbates problems.

I apologize for interrupting a productive flow of thought. But I was curious what you guys were talking about. Seems pretty esoteric. :chin:

"Would any expert academic in your field cite a Wikipedia page in a peer review article?"

Most of my publishing was done years ago, before Wikipedia, and I don't read the journals very much anymore. So I can't say with any degree of accuracy. But times are rapidly changing, and in the past I have found mistakes in reviewed articles, so I would guess that in time Wikipedia will be cited if merely to acquaint a reviewer with arcane material or the latest breakthroughs.

The entire structure of reviewing, refereeing, and dissemination should change and should put journal publishers out of business. This should be an age of open, free discussions. :chin:

for reasons is always true — TheWillowOfDarkness

In ancient Greece a bolt of lightning was reasoned to be due to Zeus. Reason evolves.

Why would an expert write a Wikipedia page? — Bartricks

Mathematics is a social or community endeavor to a large extent. And, yes, many experts from that community pay attention to what is written on their areas' Wiki pages, correcting mistakes and contributing info. The only other subject I'm familiar with is climbing, and, there, things are not quite as disciplined, but still mostly accurate.

Several years ago there was an effort to compare the accuracy of articles on the same subject appearing in Encyclopedia Britannica(online) and Wikipedia. If I recall correctly, in general Wikipedia was slightly more accurate than EB.

Secondly, the only subject that studies patterns seriously is math and I don't know if there's a subfield devoted to just the study of patterns or not. — TheMadFool

https://www.maa.org/press/periodicals/convergence/mathematics-as-the-science-of-patterns-mathematics-as-the-science-of-patterns :cool:

You would still have to follow the regulations of the wikipedia regulatory framework. Do you know its rules and how they are enforced? If not, then you are yourself not an expert on wikipedia. — alcontali

I consider wikipedia a very good source for introductory material on many math topics. If a topic is fairly popular, it is likely to be accurate. For minor topics not quite so accurate at times. The same can be said of peer reviewed articles, having been there and done that while active. :cool:

analytical continuation of a complex function has something to do with chaotic systems? Did I understand correctly? — Mephist

Dynamical systems! Actually, I was more general, showing certain sequences of linear fractional transformations can have their regions of convergence expanded by the use of fixed points. Continued fractions are a special case.

Here is a simple example illustrating the use of a fixed point:

Given $F(z)=\frac{1}{1-z}$

Then $F(z)=1+z+{{z}^{2}}+\cdots ,\text{ }\left| z \right|\lt1$

For $\beta=0$ the T-fraction expansion of the power series is

${{T}_{n}}(\beta )=1+\frac{z}{1-2z+\frac{z}{1-z+\frac{z}{1-z+\ldots +\frac{z}{1-z+\beta }}}}$

The continued fraction value is by definition

${{T}_{n}}(0)\to CF$


Here it is found that

${{T}_{n}}(0)\to F(z),\text{ }\left| z \right|\lt1$ and

${{T}_{n}}(0)\to \tfrac{1}{2},\text{ }\left| z \right|\gt1$

Whereas

${{T}_{n}}(\beta )\to F(z),\text{ }\left| z \right|\gt1,\text{ }\beta =z$

Here
$\beta =z$ is the repelling fixed point of the function $t(\zeta)=\frac{z}{1-z+\zeta }$

Someone with credentials would or would at least know what to do to test whether we've got the real-deal on our hands — Bartricks

Makes me regret not having those mysterious credentials you may or may not have that would allow you and I to debate issues that may or may not resonate with those who consider themselves authorities, but may not be. I feel defective. :cry:

I had one semester of senior-level philosophy. You?

My prof laughed at metaphysics. Although I proposed Leibniz's monads to him as a legitimate metaphysical actuality. I think it is. :smirk:

But l think there are artists who don't want their art to be public knowledge and they keep it to themselves. — Wittgenstein

And doing so, make it difficult to ascertain if they exist. But, assuming they do exist, one cannot determine whether their virtual art is really art: did they have the intention to produce art, which may or may not exist? :chin:

Do you have any formal qualifications in philosophy? — Bartricks

Please display yours. If they exist I'm not seeing much evidence. :gasp:

But then, I have almost none, so what do I know? (I can guess your reply) :sad:

Well, not only is useful, but if you find a relation between apparently completely different areas of mathematics, maybe those concepts have in some way a deeper meaning. — Mephist

Yes, I'm familiar with the notion, although I have no experience in an algebraic venue. For instance, many years ago I showed that convergence of complex limit periodic continued fractions useful as functional expansions could be accelerated by employing a feature of dynamical systems: attracting fixed points (Proceedings of the AMS). And could be analytically continued by using repelling fixed points (Mathematica Scandinavica and Proc. Royal Norwegian Soc. of Sci. & Letters). There are deeper meanings here by locating these concepts in theory of infinite compositions of complex functions.

I also showed that the traditional Tannery's theorem makes far more sense when embodied in more general infinite compositions rather than merely series and products, or integrals. Not quite what you are stating, but close. You guys are on a roll! :cool: