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:

Not totally sure what he does, something with creating new algorithms. I think he had wanted to teach originally. — Count Timothy von Icarus

Universities are increasingly using adjunct faculty, which is a travesty since the pay is extremely low and no benefits. Of course, if you are brilliant and get your doctorate from Princeton or the like there are ample opportunities. These days new faculty tend to be recruited from other math departments.

Once that's settled, is philosophy sexy? :smile:

Yes, it is a simple matter that (D) is the correct answer. — TonesInDeepFreeze

Of course. :roll:

Hello Z. I retired as professor of mathematics at a branch of a state university over twenty years ago, so its possible the undergraduate curriculum has changed a little since then. Since you belong to TPF I assume you have an inquisitive mind and are good at the math you have used. That will make your academic journey much more pleasant. What university are you attending? That could make a difference in my advice. Are you of typical age for a freshman in college, or older?

Do you have an end goal in mind? It should be easy to find a teaching job at the high school level with a BS, but an academic career at the college level much more problematic. And I don't know what opportunities there are in private businesses and government, although I suspect they exist.

I received a math BS in 1958 from a large state university, went into the USAF and studied meteorology, becoming a weather officer, then went back to school for an MA in math, taught for three years at a small college, and finally a PhD in 1971. I would have been better off academically had I gone straight through, but I was not that dedicated at the time.

You will have to take a number of courses in various areas of math, some of which you will enjoy, but others not so much. To go all the way through a doctoral program requires many more math courses, increasingly difficult. Again you would like some but not others. I only felt like a real mathematician when I reached the point where I could actually explore ideas that intrigued me and do original research. So that career path is long and hard.

So, tell us a bit more about yourself (if you are willing to do so). :cool:

Francisco Varela's homepage: F. Varela's Homepage

A bit confusing for me. The math is pretty vague with the first appearance making no sense. A little later on his discussion of dynamical systems is a tad more palatable. Overall, what he writes (in great length) could be brilliant or a satire on science. I assume the former since Stanford is behind the publication.

Peter Lynds writings make more sense to me, although he is an amateur physicist. Initially, he agreed with Bergson that time had no "instants", but rather interludes, an approach to the nature of time that drew considerable criticisms from the physics community.

To illustrate non-linear time, this graph shows the correspondence between ground observer time (horizontal axis) and proper time (vertical axis) for a spaceship going at v(t)=ct^(.1) over a time interval [0,1]. Although the clock on board ticks at a linear rate, to the ground observer a short interval in his time frame on the horizontal axis, going up to the red curve and over to the vertical axis shows a smaller interval of elapsed time on board the ship. The blue curve shows that, for a stationary spaceship, the elapsed times are the same, measured in linear form. (ignore gravity)

My response to this is that seven billion people on Earth has not hit the minimum headcount that would make my advice work by way of creating a critical mass of willingly paying customers for philosophical insight. — god must be atheist

I think this is the best contribution to this thread. Thank you, gmba.

Kind of like herd immunity in reverse.

[My math paper:

1. log24=2

Therefore...

2. The Riemann Hypothesis is true.] — Agent Smith

You funny. :cool:

Are there other non-numerical things? — Raymond

I used to teach point-set topology. A delightful topic.

I try be brief :smile:

How are math theorems found? — Raymond

I be brief. Usually by playing with existing mathematics. At first a conjecture, then comes a proof.

Can time be a variable? How do you vary time? — Raymond

Pondering

I’ve taken infinite to mean it’s always existed. — AJJ

Eternal, perhaps?

This is done to the tiniest details to avoid inroads of criticism. You explain everything, like in a math proof, leaving nothing to guesswork. — god must be atheist

Well, here's a difference between a PhD math thesis and a publishable math paper: In the thesis the grad student is encouraged to spell most arguments out in at least some detail, but a math research paper frequently glosses over any details that have relatively brief proofs and experienced mathematicians can be expected to fill in the blanks.

Which makes me wonder if this is true with philosophy papers as well?

[joke] Even your two takes on the subject are just long, convoluted ways of saying tl;dr. [/joke] — T Clark

I try be briefer in future. :snicker:

Can't believe this thread is still going on. :roll: