If I elaborate further it will sound like bullshit so I must stop here. — invicta

I applaud your perspicacity. There are mathematical ways of avoiding circles, but not now, I think. :chin:

Wiki puts SB in the category of number theory. Perhaps it's analytic number theory since limits appear. Apart from the simple continued fraction expansion it's beyond me. :cool:

SB is a fairly deep mathematical subject, as seen here.

As we get older time seems to pass more rapidly. At the age of ten one hour seemed like an eternity. Is it possible the "rate" at which time passes actually is varied? But we only have time itself to compare it with, so we cannot tell, apart from rapid motion in special relativity. As I sit typing this, is time fluctuating, and if so is the fluctuation a local phenomenon?

In a typical spacetime graph (2D space + Time) how is it that passage of time is allowed at a purely spacial stationary "event"?

Is Elon Musk really a time traveler from the year 5000AD? :chin:

In the metrics of spacetime, should time be entirely independent of spacial change? — jgill

if space is removed from time then the notion of space loses meaning I’d say. — invicta

What I was referring to is seen in the Minkowski metric of spacetime, in which the time term is in fact a distance term $c^2\left(t_1 - t_2\right)^2 - \left(x_1 - x_2\right)^2 - \left(y_1 - y_2\right)^2 - \left(z_1 - z_2\right)^2.$

Agree with the idea that it’s not time that is being experienced but change, although the two concepts of time and change are inseparable. — invicta

In the metrics of spacetime, should time be entirely independent of spacial change?

If the universe is the mathematical universe of Tegmark, then perhaps it is a giant dynamical system that goes its merry way starting from a set of initial conditions. As to the nature of God's involvement I would not dare speculate.

Are there facts about reality that will forever be beyond the comprehension of humans, like my dog being unable to understand even the elementary aspects of calculus?

I tested the Bing AI in the following way: I have a low-priority mathematics page on Wikipedia, so I asked Bing what is known of this particular subject? Now, there are a smattering of papers on the internet on this subject; what Bing supplied was the first introductory paragraphs of my webpage, word for word. That's all.

Sorry, but that's an invitation to a crazy train — TonesInDeepFreeze

I know. But his quest for the smallest particle of space is no more absurd than Tegmark's mathematical universe IMHO. :cool:

So Bernie announced he's not running, and will be supporting Joe — Mikie

Maybe they could run as a geriatric team. But which one is Gandolf?

My impression is that finite SB strings describe 'destinations' (numbers) and infinite SB strings describe 'journeys' (unending processes with no destination). My issue is that I don't see how decimals are any different. Why can't we say that (non-repeating) infinite decimals are journeys that are described by unending processes (e.g. limits) and not 'destinations' (numbers)? — keystone

I don't see what the problem is here. You can say what you want, when you want. But asking mathematicians to go along with your ideas of synatax and grammar is another matter. You tell a math person such and such is a "journey" or a "destination" and they might nod their heads and say, Well, you might say that informally. The more precise language of set theory is what they would normally speak in this intellectual environment.

I was a math professor for many years and I am not offput or disturbed by your speculations. I have encountered such notions before, many times, and have never felt discomfited by these discussions. But if I were teaching a class in foundations (thankfully, I never did) I might discuss these ideas in more detail.

Who are you trying to convince here? Philosophers who consider definitions optional?

This sort of argument is at most peripheral to most serious mathematical discussions. If you are curious about deeper, somewhat more sophisticated concerns about the foundations of the subject, look up some of @Metaphysician Undercover's posts about the ultimate nature of mathematical objects, points vs contours, infinitesimals, etc.

:up:

I worked in the analytic theory of continued fractions years ago, and one of my forebearers was Omar Khayyam, the Persian poet and mathematician ca 1100 ad . He may have devised the continued fraction expansion of the equation

${{x}^{2}}-x-1=0$ => $x=1+\frac{1}{x}=1+\frac{1}{1+\frac{1}{x}}=\cdots =1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots }}}$

Which is a lot more palatable than Stern-Brocot. Well, in my opinion. :cool:

I had thought Newsom would have a good chance, but news out of CA has not been good lately.

How do you finitely and completely describe these mathematical entities (irrational numbers)? — keystone

Pi is the ratio of circumference to diameter of a circle. The Golden Ratio can be defined as the ratio of a particular line segment to another - you can look it up on Wikipedia. Other irrationals, have at it.

I had never heard of the Stern-Brocot tree before you brought it up. But my knowledge of number theory is poor.

2024: Banana Republican Donald Trump vs Senility Candidate Joe Biden. :roll:

I can't say I recall a single revolt in history with a median age of 55, but if you look at armed protests in the US that would be my low end estimate for age. It's weird, especially since half the nation's budget is transfer payments to seniors. I suppose it is more about social control, not economic factors though. — Count Timothy von Icarus

I would like to see some statistics to this effect. That would not be my guess.

Ageism is still considered fair game, while other forms of discrimination may be declining.

Wikipedia says that the decimal representation of the Golden Ratio is 1.618033988749894...
https://en.wikipedia.org/wiki/Golden_ratio

Do you disagree with this? — keystone

This is a numerical approximation to a geometric concept, like 3.14159... approximates another ratio, pi.

Since these expansions are non-ending they do not completely describe the mathematical entities they represent. On the other hand when these entities are used in a numerical sense the tail ends of these expansions are chopped off according to the degree one wishes to approximate an answer.



:up:

The Tree gives rational approximations to the GR. There is no infinite digit RL. How would you use infinite decimal digits to define the GR?

You can call RL anything you like, but that does not mean it is so under foundation theory.

This is just a scheme associated with ratios of Fibonacci numbers and the Euclidean algorithm and continued fractions. You can call the GR the "last" RL if you like. Perhaps there are those who would agree with you. Just not in the mathematical community.

Any alternative to subscribe without going through PayPal? Got hacked in an account years ago.

I get scam claims through their site frequently.

1+1 = 2 is true in all circumstances because it’s a calculation performed on values which are simple by their numerical nature — invicta

No. Better stay away from mathematics for your "true under any circumstances" example.

A poetic and well written OP. :chin:

If 5 machines can make 5 devices in 5 minutes, that means each machine can make one device in 5 minutes. — Pierre-Normand

There is a subtlety here that GPT4 fails to address. But that's better than the other GPT.

Reasoning is a problem, as seen in the question,"If 5 machines produce 5 products in 5 minutes, how long will it take 100 machines to produce 100 products?" I'm not sure what version was asked the question, but even with coaxing and additional info it could not give the correct answer.

but whatever the case may be I don't think that my role in life is one of imitation although I could be mistaken. — Average

My late aunt used to say about my grandfather, "It took him a long time to find his niche".

Certainly, biologically, we are here to reproduce. We are fitted through evolution to survival and procreation. But most of us go beyond that, thinking,There must be a higher level purpose to our lives. I assume you are not an old person. Am I right? Because, when old we look back on our lives, thinking, What have I achieved and is it important to have achieved anything? Have I made may mark, and is that a necessity?

Or, we don't think about these things. Living life as we can. I have been an existentialist for most of my life; when I reflect, I think creating meaning has served me well. Don't ponder too long about this issue. Get movin' and find your niche. :cool:

:up:

Btw, I think Nicko got banned — Metaphysician Undercover

Yes, he did. And that is too bad. You get a scientist on the forum who "have at it" an argument using science in panpsychic realms and he's banned.

↪Largo

I was assigned to read primary texts. I came to appreciate commentary later on. But I am glad I did not start with that. — Paine

I had the opposite advice in my one (senior level) course back in 1958: Read commentaries first, then primary sources. I had tried to understand a particular philosopher in order to write a report, but flamed out there with little understanding of what he was writing about. The professor then told me to go to commentaries at first - which worked.

My name for Christian culturist is "Jesus fans." The don't actually follow his teaching, but they say he's a really great guy. — Art48

George W. Bush when asked who was his favorite philosopher answered, "Jesus Christ". Then, I thought, How ridiculous. Now, not so much.

But philosophical practice and philosophical writing are not the same. The ancient practice of philosophy was not about writing but a way of living. — Fooloso4

Is it possible some philosophers when writing run out of ideas, but continue writing? :chin:

If something cannot be experienced and cannot be exactly defined, then we should not speculate about it. — ClayG

Does that mean that a great many if not most threads on TPF should be removed? It's already been decided that definitions here need not be exact - or, in some cases, comprehensible.

I don't see how there are any issues of incomprehensibility that are not ultimately an issue of length, — Ø implies everything

Quite possibly. I don't know. Sometimes it may seem to take forever to get to a point of understanding a complicated idea or general area of study. Which causes me to wonder if AI may move beyond our comprehension of mathematics it devises - length or not. Here is an interesting introduction to a paper on Scheme theory - a topic beyond me, I fear:

Scheme theory, perhaps more than any other subject, has a reputation for being extremely
difficult and tedious to learn. One gets the impression that the subject involves many highly
technical and difficult constructions, is exceedingly vast and abstract, and that it takes
considerable time and energy before one is able to prove anything of value. Quite famously,
the subject originated from Grothendieck’s attempt to “simplify” an eighty page paper by
Serre into the thousand page document that was to become Les ́El ́ements de g ́eom ́etrie
alg ́ebrique — a fact that is both oddly remarkable and offers little encouragement.
It is perhaps somewhat surprising, then, that there seems to be no shortage of graduate
students and even undergraduates eager to devote time to understand schemes. The usual
procedure is to sit down with a copy of Hartshorne, formally sift through a seemingly endless
series of complex definitions, and then grudgingly admit defeat. Usually absent from these
attempts at understanding schemes are good sources of intuition, motivation, and clear and
identifiable goals. The result is that students learning the subject this way have difficulty
explaining the “point” of a definition or a construction, and so don’t know what it’s related
to, why it’s there, and consequently can’t use it.
The purpose of this article is to give the basic definitions of scheme theory in context.
We will take the view that it is just as important, if not more so, to explain the definitions
themselves as it is to explain the lemmas and the proofs. In doing so, we hope to remedy a
common affliction that befalls those who read Hartshorne’s book: not having any idea what
is going on

David Urbanik

Brilliant OP. I would expect nothing less. :clap:

Could you elaborate? Do you mean their theorems/concepts would have so many steps/components that it would take longer than a lifetime for a human to properly go through it/grasp it? — Ø implies everything

You mention one possibility. Yes. And the sheer breadth and size of the subject is overwhelming, almost impossible to keep up with, having over 25,000 pages on math on Wikipedia. Also, 200-300 research papers a day, every day, on ArXive.org . Then we have things like the Four Color theorem which required a computer to evaluate a huge number of cases to "prove". By themselves, humans can only process a limited amount of interwoven logical steps. Then there are probably limitations on the size, scope and complexity of new math concepts humans can conjure up. So mathematicians with computers may enhance the realm of the subject enormously, but if Tegmark is right the entire universe is somehow mathematical and probably beyond comprehension. Being all-invasive, this math structure would be conscious in some way, sending out trivial ideas to people calling themselves mathematicians. :cool:

Max Tegmark's Mathematical Universe ideas might relate to this. I think it is only a matter of time before AI fabricates concepts, techniques, theorems and proofs in mathematics that lay beyond the grasp of human mathematicians. There is your higher intelligence, embedded in a universe whose Matrix is more akin to Hilbert Spaces than what is seen in the movie.

I want not a life without wants. But I want not to have too many wants. It's a matter of balance.

I will be unhelpful and say it depends . . . . . — Count Timothy von Icarus

Yours is among the very best posts I have read on TPF. :clap:

Sensations are nervous system-dependent — 180 Proof

:up: This sums it up pretty well unless you believe rocks have feelings.

Do you think there is a math brain or a type of person to whom math speaks? — Tom Storm

Genetics are a big part. A little like musical talent. My father grew up in a poor coal mining community in Pennsylvania where almost all the young men went into the mines after high school. He worked after school in his senior year in the mangers for donkeys underground, doing his homework by lantern light. But he escaped his origins and became a professor of business statistics and directed the grad program at the University of Georgia for a while. He had a masters in mathematics, then a PhD in statistics. My degree was in math.

People who get stuck on specific definitions are often irritating pedants and seem to miss the point. — Tom Storm

Guilty. Just my math background showing. :yikes: And the old adage from CS: "garbage in = garbage out".

But, as @fdrake explained, definitions in math sometimes undergo revisions as the process of exploration or problem-solving progresses. Research in math is a very fluid work space if one is not forced to make it advantageous in some applied problem. It can be remarkably loose, going back and forth. I'm looking at something now that will necessitate a revision of hypothesis - the starting points, like definitions, for successful arguments.

Although it irritates me at times to read sloppy, ill- formulated definitions in these philosophical discussions, I am growing to understand it's part of the process that might converge to an interesting conclusion.

It's a little hazardous to form an analogy outside physics with a concept or result in physics that one does not fully understand. Especially when numbers are involved. But bully for you to give it a try. No cigar, however. :roll: