You said a thing is flat to the degree that it's not curved, and a thing is curved to the degree that it's not flat — Metaphysician Undercover

Theorema egregium.

The "remarkable", and surprising, feature of this theorem is that although the definition of the Gaussian curvature of a surface S in R3 certainly depends on the way in which the surface is located in space, the end result, the Gaussian curvature itself, is determined by the intrinsic metric of the surface without any further reference to the ambient space: it is an intrinsic invariant

(Wiki)

Did residues and contour integration, etc. but this seems new. Or is it a normal thing in the field. Did you turn it into a visual? — Haglund

Coupled Differential Equations

There is a tremendous amount of complex variable theory that is not covered in a typical course (which I taught a number of times). I got into my specialty over a half century ago via complex valued continued fractions, which led to a general study of infinite compositions of complex functions.

Yes, I extracted the visuals.

At your level you are basically learning about tools and preparation. It has nothing to do with mathematics — I like sushi

In fact, as fundamental tools it has everything to do with mathematics. The algebra and trigonometry and perhaps basic calculus you learn is useful throughout the incredibly diverse realm of mathematics.

G'nite my friend. :yawn:

Did you succeed with that Victorian cross (no offense!)? — Haglund

Not more than a fraction of a second. It awaited a smaller, stronger body.

If self creation is coherent, then there can be nothing and then something.
That isn't something from nothing. That's nothing and then something. The cause of teh [the] something is not the nothing, but the something itself. — Bartricks

Moments of clarity like this make a mockery of claims that all that is worthwhile has been mined from philosophy. :chin:

What's a toity world? Well, it's just a device to make clearer what a toity truth is. — Bartricks

It's refreshing that, having assumed all avenues of original philosophical exploration had been traveled, something novel and appealing bubbles up from spilled beer at the local bar. :strong:

I guess it pales in the face of your accident about 35 years ago... — Haglund

Sadly that's the strongest part of my body now. :worry:

↪jgill
Looks very Mayan! Perhaps its just the combination of color and the swirling motif. — Agent Smith

Hidden within those intricate folds are the secrets of the universe.

Well, maybe not. :sad:

Not so. He was an officer on the front lines, decorated several times. — jgill

Thanks. I did not know that. — god must be atheist

I knew Lester Germer slightly, as a famous physicist (wave/particle duality) but more as a fellow rock climber. My impression of him grew considerably when I found he had been a fighter pilot in WWI. A multi-dimensional person.

That was more Kripke — Banno

In Philosophical Investigations §201a Wittgenstein explicitly states the rule-following paradox: "This was our paradox: no course of action could be determined by a rule, because any course of action can be made out to accord with the rule"

Yes, the bizarre addition example.

He was a complete fuck-up. — god must be atheist

Not so. He was an officer on the front lines, decorated several times.

(But I kind of agree regarding his wondrous rule-following paradox.)

Begin with a number between 0 and 1 in the first cell. Next cell = previous cell x (1 - previous cell) x some constant between 2 and 4. — Cuthbert

Sound like elementary cellular automata, championed by Wolfram some time ago. I've written programs that show a constant sort of development, then a jump to a weird line or something. Wolfram thought he had come upon a hugely important concept, writing a book with over a thousand pages. I, like most other readers, gave up after a few hundred pages.

Somehow they seem to eat each other. Raw sex in the complex plane... — Haglund

Like Pac-Men. My friend, you need to get out of the house more often. :wink:

But I often find the talk pages on Wikipedia more informative than the articles themselves, especially on philosophical subjects — magritte

I do also. In this case [mathematical physics] the talk section makes one's head spin!

Where that presents difficulties, is that there is no provision in most people's minds for things to exist in different ways — Wayfarer

If things were not bizarre enough: Rydberg polaritons

I'm tied to my home for a still few weeks — Haglund

You live in Shanghai ? :smile:

So philosophy's not your cup of tea — frank

More a form of entertainment. :cool:

Have a good evening — frank

You too. :smile:

Laplace's demon has been upgraded with the latest software by David Chalmers — frank

The guy who believes rocks have feelings?

I've often wondered how the aether affects ectoplasm.

Wiki:

Laplace's demon was based on the premise of reversibility and classical mechanics; however, Ulanowicz points out that many thermodynamic processes are irreversible.

Gotta move up to the 21st century, buddy.

Random numbers are generated by a deterministic system. In a computer it's a quartz oscillator — frank

Suppose my random number comes from an observation of unpredictable minute changes in atmospheric pressure?

Have you written some background on the maths involved? — Banno

Lots of notes/articles as a hobby. Here's an example: Woven Contours

I've written all my computer programs in BASIC.

You have an algorithm that, once begun, leads to an outcome - thus determined. But halfway through the algorithm is a step requiring the input of a random integer between 0 and 9. Is the outcome determined? Random? Both? Neither?

Can you zoom in like in those colored fractal zoomings (where the colors represent a rate of convergence, if Im not mistaken)? — Haglund

It's not a fractal, but sometimes one can focus on a spot and enlarge it and find additional intricacies. Iterations are done at pixel levels with light shades when the modulus is great and dark shades when it is small. Very simple.

↪jgill
Impressive. This is yours? — Banno

Yes, I've done lots of unpredictable images. Look at my icon on TPF. Here's another I call Reproductive Universe:

Here is a non-chaotic image arising from an iteration process that involves differing functions rather than a single function. Its' beautiful intricacies might possibly be predictable were an attempt made to do so, but it would take a significant effort. This is a dynamical system in C that rewards its inventor. I call it Dream of Gold.

To Mathematicians

Is Chaos Theory (math) an admission that the calculations involved are too complex for humans and current top-of-the-line supercomputers (extremely difficult to predict) or is the claim that there's true randomness (unpredictability). — Agent Smith

The computer calculations define the dynamical system in C (complex plane). Iteration of a function carries an initial point in C to a new position - usually not the same position unless the original point is a fixed point of the function. When there is a condition in which two points very close to one another diverge dramatically and relatively unpredictably under iteration the system may be chaotic (under other circumstances fractals might appear). This might occur everywhere on a set S in C, or on a part of S. This is the simplest version of chaos theory. Draw your own conclusions. :cool:

Calculus, complex numbers and chaos theory were developed to cope with the ineffectiveness of current maths to deal with emerging problems in physics — Cuthbert

Not entirely true. Complex numbers arose in the study of roots of polynomial equations, more pure math than physics.

It’s simple: you can’t step even once because, as soon as you touch the water, one instant later it is not anymore the same you touched initially, because it is flowing. — Angelo Cannata

The word "same" means persistence over a span of time in this case. The instant (t=0) your toe touches the flowing water all is frozen in time. Think of a photo taken at that instant. For t>0 the word same comes into play. This becomes an argument of time as instances vs time as periods. Think Einstein vs Bergson (1922).

It is similar to Zeno’s paradox of the arrow, but the opposite way. — Angelo Cannata

Nice point.

Mathematical physics are dynamical systems where anything that is mathematically possible is also physically possible until the theory is shown to violate some physical law. — magritte

Not really: Mathematical physics. A person engaged in this pursuit seeks mathematical ideas and procedures that might illuminate aspects of physics. Sometimes new math is developed in this quest, but there are over 26K separate topics in mathematics on Wikipedia so there is a wealth of material one might search for applications in physics. Also, mathematical physicists attempt to put foundations in place to prove certain math procedures in physics - like renormalizations and path integrals - are mathematically sound. Dynamical systems, a very specific area of mathematics, sometimes are referred to as classical mathematical physics.

Your position sounds similar to the ancient philosopher Cratylus, "you cannot step in the same river once." — Jackson

Sure you can. But not twice. :roll:

A minor point, but when a mathematician explores a possible theorem, searching for a hypothesis that will guarantee a certain outcome, he does so over a period of time. But when he finds such a starting point, the conclusion instantly exists. A implies B involves no time component. Is this true of cause and effect in the physical world? Hitting the ball with your bat is certainly cause and effect, possible only during an interval of time.

Here's the first article I've seen that discusses the possibility of determining whether alternate universes might exist. It still seems a reach.

In mathematics, a dynamical system might proceed to evolve along alternate paths at points of bifurcation. But what happens in math may be mere fiction in the physical world.

I find a certain merit in your arguments. By making philosophy equivalent (?) to science hypotheses you have put the subject on a firm foundation, distancing it from the reams of babble that might now be quartered in the bin of metaphysics. If I understand your position. :chin:

Philosophical exploration in science is known as Scientific hypotheses — Nickolasgaspar

How would you categorize the many worlds interpretation of QM? At this stage it is not a scientific hypothesis since testing it is a distant objective. Is it thus excluded from being a "philosophical exploration" in science? Is it metaphysics?

Philosophical exploration in science is known as Scientific hypotheses. Those hypotheses need to be testable in order to be scientific. — Nickolasgaspar

By virtue of them being labeled scientific hypotheses they are testable. Philosophical exploration might be any sort of babble. Quantum mysticism, etc.

Mathematics are NOT science. Its a tool based(that science uses) on an accurate language of logic that has the same role like human language in Philosophy. — Nickolasgaspar

And there's the problem. Human language lacks the preciseness of mathematics. Start at ground zero: Ontology. "the nature of being"? "being" being what? Existence? Human existence, physical existence Platonic existence, of mathematical theory, of consciousness, of memories, etc.?

Math has its confusing moments also. Category theory is one for me! But, typically, a mathematical argument is a model of clarity, frequently concise and compact, compared with the tsunami of words constituting a philosophical argument.

You can not do philosophy without having basic empirical observations to start with . First we interact empirically with your environment, we form our philosophical questions and hypotheses and we look back at nature for additional information that could provide answers and validate some of our hypothesis. — Nickolasgaspar

Philosophy of mathematics. Foundations of mathematics? Perhaps the formations of logical principles by observations of natural phenomena.

You can NOT have science without philosophy and philosophy without science — Nickolasgaspar

My interpretation: Philosophical explorations in science are speculations in science. When they are not done by those well-versed in a science they rarely have any intellectual depth.

Thus we differ. That's OK.

Metaphysical views ....are metaphysics — Nickolasgaspar

I agree. :chin:

I see your icon only lists physics as a part of philosophy. Are you saying all the other branches of science are spin-offs? Or is this icon from the distant past?

What is "Total Apochavnosis"?

Again ALL scientific hypotheses are Metaphysics. Mathematics are NOT science. Its a tool based(that science uses) on an accurate language of logic that has the same role like human language in Philosophy. — Nickolasgaspar

Thanks for the heads-up about math. :cool:

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices

(SEP)

So even tools can be metaphysical.

Philosophy is universal phenomenological ontology — Xtrix

Well, that may clarify things. No need to continue. :yawn:

What do you think philosophy is? — Nickolasgaspar

After 426 posts on this thread there doesn't seem to be a consensus. It's certainly not the philosophy of nature of past ages, having been eclipsed by modern science.

Science isn't interested in what Big Bang means for humans — Nickolasgaspar

In fact, astrophysicists study the results of the BB and what they find may very well have implications on future space travel for humans. But I hear what you are saying.

So can we agree that Science is Philosophy that doesn't deal with meaning and value because those doesn't have objective metrics? — Nickolasgaspar

Science is full of meanings and values that are germane to discussions within those disciplines. It's usually the scientists who participate. Whether they perform "philosophy" when doing so seems irrelevant.

How about the idea that metaphysics is the condition of possibility for understanding the theoretical framework within which proven facts make sense in the first place? — Joshs

So metaphysics is foundational, without which interpreting scientific results and speculations go nowhere? This is a bit vague for me, but my background in philosophy is limited. When I think of metaphysics I think of infinitesimals. I need specific examples of concepts in order to understand the concepts. I don't think of string theory as metaphysical, although it may be seen as a result of an initial metaphysical trigger.

The conclusion is that science has never ceased being ‘philosophical’ in the sense that theoretical frameworks represent a naive metaphysics. — Joshs

I agree if pondering implies philosophical.