We learn by observing nature. Then we take those observations and extract their essences. — jgill

I'm with you in spirit, but perhaps we dream up those essences and only later learn to check if they or their implications are compatible with observations. — igjugarjuk

In all my years as a mathematician, however, I must confess that I have never worked in applied mathematics. Like most in my profession, I explored an intriguing abstract concept. Still do.

At that point I think we start to take philosophy less seriously — ZzzoneiroCosm

You mean there are those who do take it seriously? :snicker:

It's interesting that the expression, metamathematics , coined by Hilbert a century ago, is much easier to describe than the more archaic metaphysics. Discussions of the former rarely use that term, whereas the latter produces volumes, and still gets nowhere it seems.

Sartre's notion of pre-reflective consciousness as nihilation. This was absolutely new. Historically, no philosopher, other than Sartre, came up with this notion.

I anticipate that there will be many other such future geniuses — charles ferraro

Perhaps you could summarize this idea of Sartre's briefly.

The word metaphysics should be eliminated from discussions of science. It is archaic, like a lady two hundred years ago suffering from the vapors after witnessing a strand of ectoplasm.

In science,metaphysics is an archaic word replaced by speculation in science.

At first glance one might conclude that the masters of generations long past did it all due to the ongoing discussions about those philosophers. On the other hand there are areas like the philosophy of mathematics and the philosophy of physics that are vibrant.

Look at all the babble about quantum theory. And judicial philosophy easily reaches the common citizen with reports of deliberations by the SCOTUS and actions and inactions of the Attorney General.

How can ethics not be on a top burner with the ongoing war in Ukraine, and a philosophy of international relations.

Dynamical systems (causal) can be so complicated that stochastic-correlational is the only way to work with them. Pretty much all it takes are functions that are not invertible.

The computer could be, for example, a huge cellular automaton — Gnomon

Wolfram (creator of Mathematica) attempted to convince the scientific community that cellular automata were at the heart of virtually everything physical. He failed.

As a mathematician who never gave much thought to Platonic ideals, my rather superficial view is that these ideals do not exist in any sort of physical forms, but exist in an abstract space that is accessible to human minds, in much the same way that spaces of functions exist in the normal mathematical realm. Abstractions from reality are commonplace in math.

It seems as if we are more satisfied when we can picture things than when we are left with no choice than write equations — Manuel

You and me both. The math I explore has strong visual components, but not from nature. Patterns in the complex plane mostly. Look at my icon.

:up:

I'm surprised your PhD program didn't have this as a requirement. But, like math, there are many paths to follow.

So much confusion about Quantum theory occurs because those who are not physicists try to take mathematical descriptions (they probably do not understand) and turn them into physical objects.

The wave function is a probability amplitude form that has complex values - not something traveling through space. Superposition arises from a linear differential equation (DE) having linear combinations of solutions, not some mystical process hovering in the aether ready to appear after some sort of magical "collapse".

The concept of "measurement", which we all assume is something like an electronic ruler or scale, seems to be at the heart of understanding QM and has no easily understood definition. Wiki:

In quantum mechanics, each physical system is associated with a Hilbert space, each element of which is a wave function that represents a possible state of the physical system. The approach codified by John von Neumann represents a measurement upon a physical system by a self-adjoint operator on that Hilbert space termed an "observable"

As I have pointed out before, a stripped down version of the Schrödinger equation is nothing more than a very simple DE describing, for example, continuous compounding from your local bank. Somewhere in all of this is the miasma of entanglement, which only God seems able to unravel. :roll:

Sorry, but I think such speculations are ridiculous.

We learn by observing nature. Then we take those observations and extract their essences.

The language he [Wigner] uses is deeply religious throughout — Moliere

It's funny you say that - his Wikipedia page says he was a convinced atheist — Wayfarer

Interesting observations. My ex-wife's father, a friend of Wigners, was an architect and intellectual in Hungarian society. He had no use for organized religion, but was something of a disciple of Tielhard de Chardin, an intellectual and Catholic priest who advanced the idea of an Omega Point, toward which the world moves and reaches in its final days. A curious blend of science and something like religion.

First off. How did things get this out of hand? — TiredThinker

Bat Masterson and others with their tales of the Wild West. Every generation having veterans of a previous war. ???

Measurement doesn't affect anything in the past. — Andrew M


It does! It collapses wave function in space and time. The past was in superposition untill we measured it. Remember Copenhagen... — Landoma1

Welcome back, quantum mysticism. "Collapse of the wave function!" carries us away from Earthly distractions into the cirque of the gods where ectoplasm interacts with aether causing spacetime curvature. Superposition is annihilated with a bolt from Zeus!

It's a mystery to me. All I know is we mathematicians observe physical phenomena and extract and abstract patterns.

↪jgill

There are many branches of metaphysics not just the one. Perhaps some metaphysics as a philosophy of mathematical fundamentals done by mathematicians might be illustrative. — magritte

Of course there are. I've long considered infinitesimals metaphysical objects within mathematics. Transfinite set theory seems metaphysical to me. Like pornography, I know it when I see it.

Not knowing the existence of unknowns can trip one up.

Also, what fixed point do you have in mind? — TonesInDeepFreeze

This caught my attention although I am not following the discussion, not in my bailiwick. I've worked with these things quite a bit in Banach spaces, etc. but not in this context. What's going on?

I never read posts about wiki — Jackson

I understand. Don't read this one.

You could say that of logic — Jackson

Logic is the study of correct reasoning or good arguments. It is often defined in a more narrow sense as the science of deductively valid inferences or of logical truths. In this sense, it is equivalent to formal logic and constitutes a formal science investigating how conclusions follow from premises in a topic-neutral way or which propositions are true only in virtue of the logical vocabulary they contain

Wiki

It is a branch of philosophy like epistemology or ethics — Jackson

Then it should discipline itself by agreeing upon a definition.

What does the discipline of metphysics have to do with quantum entanglement? — Jackson

Calling metaphysics a "discipline" is quite a stretch. Here, on TPF, several philosophers have tried to interpret QE without a clue about the math that seems to support the phenomenon. That's an example of metaphysics as I see it.

When does metaphysical border on mystical? Eliminating metaphysics (which I consider impossible without severely crippling philosophy) would remove all the entertaining babble about quantum entanglement, for example.

The word metaphysical itself is ill-defined. This is a step further into the void.

Acceleration has never been adequately understood by human beings. — Metaphysician Undercover

From past posts I assume you refer to instantaneous acceleration.

paradoxes; they're an existential threat — Agent Smith

That's a paradox itself. :cool:

I'm curious what you think of Schaums Outline of Logic. I found math outlines that worked well in several courses I taught. Would you recommend this for students such as ToothyMaw? I've never looked at a copy.

Wagner, not wigner

Good question. I was much younger and I think an associate professor at the time, and my interests then might not have been my interests now - hard to recall. But probably I would have discussed the origins of mathematical concepts, those ideas that seem to pop out of nowhere and can become so important. Without such breakthroughs math would stagnate. An example is the notion of metric spaces, an abstraction or generalization of Euclidean distance.

I missed a golden opportunity years ago to exchange letters with Wigner or talk with him. My ex-wife's father, who lived in Montana, was a Hungarian aristocrat who corresponded regularly with Wigner. I could have discussed this very issue, being an academic math person. Sometimes we skirt by greatness without recognizing an opportunity.

From Wigner's paper:

The principal emphasis is on the invention of concepts

, and he goes on to observe that we would run out of interesting theorems were it not for the creation or discovery of new concepts.

The basic observation piece being subject to probability seems crucial. Once the superposition is resolved it would seem that it is like the coin split down the middle to a head half and a tail half. But it's got to be more than that. Wish I had studied QM years ago.

My best friend, a fellow math prof, departed a physics major after he enrolled in QM.

The statement of a theorem is a way of understanding what has been proved. In modern abstract mathematics, however, one would need advanced knowledge to understand even that statement.

It is possible for the researcher to argue step by step, building a logical edifice, without stating the theorem. Then stating the formal structure of the theorem by say, "Thus we find that . . . ". I just did that in a math note, in fact:

"
A Brief Analysis . . ." (page 2)

What if it applied conversely and allowed us to determine that the proof behind a principle or theorem assumed to be true couldn't be observed or determined? — ToothyMaw

If a theorem is "true", then it is so subject to foundational axioms. Changing the axioms could change the designation of "true". "True" means derivable in this way. "True" isn't out there all by itself.

Well, yes. If God spoke to me saying, THIS THEOREM IS TRUE, then that would remove a nagging uncertainty in the proof process. But it might end up making little difference in the future effort. On the other hand, if He said, DON"T WASTE YOUR TIME, I wouldn't. :cool:

Well, I might think, I'll use mathematical induction and see where that goes. And it might work, or not. I can essentially do a simple proof in BASIC and see all the constituent parts, and I might have thought of the various parts before designing my program. But to know for certain the following 10 parts will be used before I begin experimenting with them is a stretch. And a rather foolish and non-productive expenditure of energy.