What about:

$\sin (x)\approx x,\text{ }\sin (x)\cong x\text{ }for\text{ }x\approx 0$

or the more mysterious $\operatorname{s}(n)\sim t(n)\text{ for large }n$

???

The latter is the least well-defined symbol. The former means approximately, which of course requires clarity.

Once one moves beyond 3D, higher dimensional "space" for me becomes an algebraic geometric concept rather than a reality. Very useful for predictions but that doesn't imply it truly exists.

The universe as a dynamical system certainly contains those critical points showing SDIC that create butterfly effects. At such points, in the continuum of time, the universe splits, with one trajectory prevailing. What of the alternate trajectories? Do they exist in alternate worlds, or simply as ghost-like speculations? Memories of non-events?

Tegmark is an interesting guy, but his ideas read like science fantasy.

As an ex-gymnast and climber, what you speak of is allowing the body to do what it does with no interference from other aspects of the mind. No thinking, just flow with the pattern, that pattern having become instinct with practice.

Intellectual activities are more difficult to analyze in this way. When I'm trying to solve a math problem I think about it to the exclusion of distractions, hours seeming like minutes. However, there is yet another dimension to this topic. As Einstein noted, taking one's mind off the subject and relaxing doing something else may allow the subconscious to send forth a solution or a new way of approaching the problem.

This effect is what hunts the philosophers of science. They are so caught up in inventing schemes and methodologies of science that they loose sight on its actual workings and are completely paralyzed when they actually engage in it. — Cornwell1

Are you speaking of philosophers of science or scientists themselves? Of course, frequently they are one and the same.

The relative God exists alright! — Agent Smith

The best among us is but human, although I would agree that Tom Brady is a God.

To equate the far end of a normal curve to a celestial being is beyond philosophy IMHO and a fantasy.

How can such ingenious structure exist by itself? — Cornwell1

Ours is not to wonder why, ours is but to do and die.

Geothermal energy does seem to have tremendous potential. There are about 100K people employed in the industry now according to Wiki. It's estimated that it would be viable as a primary source if customers were willing to pay a little more for energy. But in these times of massive inflation that's problematic.

And what's the difference with ≡, "identical to"? — Cornwell1

$2x=3-4x$
Conditional equation, true only for select values of x

${{\left( x+1 \right)}^{2}}\equiv {{x}^{2}}+2x+1$
Identity, true for all or nearly all values of x

The latter isn't used much anymore.

Perhaphs jgill, can offer better insight here, than I. — universeness

Nope. You did a great job, buddy :cool:

So I've decided to try and dip my toe in the water with programming. — Wayfarer

Good luck, my friend. :smile:

I'm not a scholar of Augustine's works, but I've yet to read anything that he wrote about philosophy that I would consider nonsense. — Wayfarer

"The good Christian should beware the mathematician and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of hell."

Poor old jgill. — Banno

Look upon my Icon and tremble, all ye of TPF !!

This thread has become nothing more than a conflation between verb and noun of the word "change"

I could find no code example — universeness

O0ps, sorry, I meant the icon is a product in the complex plane of a BASIC program I wrote . In this instance, the program created the unexpected demon from a coupled pair of differential equations: dz/dt=f(w,t) and dw/dt=g(z,t),where the functions involved contained sines and cosines.

During the time I worked I knew a number of mathematicians who would have little to do with computers. A bit surprising since numerical analysis was popular then, and some of them were actually researching in that topic. I did too, but computers were primitive and numerical analysis sought to speed up computations, even if the mathematician couldn't speak the CS language.

My initial encounter with computers was a graduate math course in numerical analysis taken in 1962. We wrote short programs, turned them in to someone behind a window where IBM cards would be punched, and finally after a day or so, run through a machine the size of a large room. Then we would find we had made a mistake, and would repeat the process over several days.

It was not a pleasant experience.

I know of some prominent philosophers of physics with physics or math degrees: David Albert, David Wallace, Huw Price, Dennis Dieks. — SophistiCat

Certainly doctorates in physics or math - especially theoretical physics - can place one in the borderlands of philosophy and science. That was my point. Thanks for the info.

I wrote many programs in my very early days as a teacher in BBC BASIC.
Having to number every code line was fun eh? — universeness

N=1/N was a trick question. Sorry :cool:

I now use Liberty Basic. I used Virtual basic until one morning I opened my computer and discovered that Microsoft had deleted the language. Over the years I have tried a number of languages, Pascal, Fortran, Mathematica, C++, etc. But it seems the more sophisticated they are, the more they cater to the popular applications in math. My interests are about as as far from popular as one can be.

I enjoy the challenge of programming a complicated and unusual math process from scratch. Click on my image to see an example.

I haven't had to use required numbering in some time. Neither VB nor LB require them.

Now how do I explain it? — Agent Smith

$\sum\limits_{k=1}^{n}{\left( 2k-1 \right)}=2\sum\limits_{k=1}^{n}{k}-n=2\frac{n(n+1)}{2}-n={{n}^{2}}$

Nighty nite, my friend

This thread illuminates what it is to be a philosopher in modern times — jgill

No, it doesn't. To see what philosophy in modern times looks like, read some actual philosophy, e.g. here: https://philpapers.org/browse/time/ — SophistiCat

Since we are talking about time, glancing over some of the papers in the metaphysics of time (the truly philosophical area) much concerns the A-theory and the B-theory, and reading on one finds that subjects in physics, like relativity theory, come into play. Here the philosopher can only use popular versions of physics phenomena in their arguments - unless they have more in-depth knowledge of physics.

In the philosophy of mathematics it would appear that one becomes equally versed in foundation and set theory in order to make contributions. But I speculate. You may be right.

It seems like some people don't consider that less can be more — TiredThinker

Yes, that's true. But you don't have to read it all. It's good that one can write at length without a referee stepping in and limiting your flow. But it might help if a lengthy piece had a short concise summary.

I mean as an explanation for where everything came from (such as Lawrence Krauss proposes) — Down The Rabbit Hole

In Wikipedia he is quoted as saying, "Turtles all the way down" Has he gone beyond this view? I haven't read anything by him.

Suppose I saw the ten first members of the pattern of the outcomes of your products. All smaller then e. Are we sure that only your prescription for generating the numbers (2, etc.) is unique? — Cornwell1

Good point. Fortunately that's not an issue normally in a mathematical discussion. Of course there may be other "rules" generating that finite sequence. For the most part in math research one generates a sequence of numbers according to a given process, and then tries to ascertain whether an observed outcome is true for all numbers so generated. I'm happy it doesn't go the other way around!

I'm reading this elementary textbook on math. The only way one can determine the persistence of a pattern i.e. know that a pattern you discovered will continue (is the correct one) is if you can explain it. — Agent Smith

Prove it continues. Mathematical induction is one way. Providing an algorithm is another. For example, I am working on a theorem now that has the product

$\prod\limits_{k=1}^{n}{\left( 1+\frac{k}{{{n}^{2}}} \right)}$

And its fairly simple to prove that, for all positive integers n,


$\prod\limits_{k=1}^{n}{\left( 1+\frac{k}{{{n}^{2}}} \right)}\le e$

I noticed that N=1/N seems to only have the solution N=1, is this the only solution? — universeness

Certainly n=1/n has solutions n=1 and n=-1. The other meaning I have in mind is quite different. Hint: I have written hundreds of mathematical programs in BASIC. :cool:

An argument against using pattern sequences in IQ tests.

N=1/N has two very different meanings in practice. Context means everything.

I would venture to guess that in BOTH cases it is not necessary to spell out existing knowledge, as long as one properly references the source. — god must be atheist

Clearly references suffice for existing knowledge. I'm speaking of steps in proofs that can be sketched out, knowing that experienced readers can fill in those steps.

Most things in the universe have no corresponding mathematical structure — Cornwell1

Perhaps they do and we are incapable of understanding them.

Only approximations will do — Cornwell1

All of life are approximations. A right triangle is a Platonic ideal, but its approximations in the physical world have been overwhelmingly of benefit to humankind.

This thread illuminates what it is to be a philosopher in modern times when all the sciences have progressed to the point that philosophical inquiries in them mean scientific speculation - requiring depths of knowledge in the subjects. Legal and moral and ethical and political issues also require specialized knowledge. What is left?

I see this in mathematics. It's hard to reach and survive on the leading edges, no matter what intellectual direction one takes. What's left to ponder are elemental concepts, long eclipsed by experts, as we've seen on this forum.

In Banno's example of color change, that change is of a dy/dx type. But as the eye moves from left to right, that change is of the dy/dt type. When x=x(t), dy/dt=(dy/dx)(dx/dt).

The study of the fundamental nature of knowledge, reality, and existence, especially when considered as an academic discipline.

Hope that sheds some light on things going forward. — Garrett Travers

Barely a glimmer.

Wiki:

Historically, philosophy encompassed all bodies of knowledge and a practitioner was known as a philosopher

Those were the good old days. :halo:

When I was six years old, I was reading a book about mathematics . . . — pfirefry

:lol:

It would seem another physicist has entered the room :clap:

Sorry for asking, but are you the "mountaineer"? — Cornwell1

Climber :cool:

but his basic notion that Reality is fundamentally mathematical makes sense to me, especially in light of Quantum Physics, where the structure of reality is a mathematical Field — Gnomon

That idea seems to predict well, but it models phenomena that are poorly understood.

:up:

My husband, a meteorologist at the NWS, once explained me. The butterfly effect is no real effect — Cornwell1

Thanks for chiming in. I was a meteorologist for the USAF many years ago so its good to hear another give their opinions on TPF.

What I'm really concerned about is if climate action has a deadline to meet and whether we're already past that date with destiny — Agent Smith

I'd guess, yes. But Cornwell1's husband is more reliable.

A function defined in set theory as a certain kind of collection of ordered pairs - not necessarily real or complex numbers - might be connected to this topic. Other than that I've demonstrated that in a general iterative sense a functional has this property since a functional operates on a function and produces a real or complex number, not another function.

It's a waste of time.

Second, is there any difference between weather & climate models used for making global warming predictions that would mean chaos theory is inapplicable to climate? — Agent Smith

"Chaos theory" refers to certain dynamical systems having weaknesses allowing for minute changes to produce significant results through sequences of iterations. Meteorology includes both weather and climate. Meteorological models are susceptible. Not all models are unpredictably chaotic. It's a sort of vague expression in everyday use.

First off, you really don't know if those beautiful butterflies are not behind the recent spate of extreme weather events. — Agent Smith

Of course I do! Just trust me. :brow:

There's a wonderful isomorphism between mathematical thought and the physical world. To go beyond that is metaphysical hogwash. IMHO. :roll:

we should exercise caution for the simple reason that chaos theory implies that even the tiniest variation in the inputs (possible in the real world) would nullify any predictions whatsoever. — Agent Smith

Not necessarily. Millions of butterflies wiggle their wings in the Orient with no effect in San Francisco at all. Not all weather models are subject to the "sensitive dependence on initial conditions" of chaos theory.

Can change be mathematically described without reference to time? — HKpinsky

Sure. A derivative can describe a rate of change with regard to a non-time variable: dy/dx

Well, to tell you the truth, I don't get it — Agent Smith

Heh, Heh. :wink:

There are instants of time embedded in infinitesimal intervals.

Certainly, I go to Salisbury University, a smaller school in Maryland, USA, and I am Junior - age 21. Plan on attending University of Baltimore Law School in Fall of 24.' The Space Force interested me as an alternative. Thanks a lot for the advice — Zolenskify

I had a friend my age in the Air Force who had a BS in math, and he was an intelligence officer who planned to go to law school after his tour. He did and was quite successful I recall. (Math 215 might be a good option for a law career) :cool: