I've read that complex numbers are a convenience . . . — Art48

True enough, but the value of complex analysis lies far more in the concept of holomorphic (or analytic) functions, where differentiation is over areas and not simply points in the complex plane. For example, when studying work one hopes for conservative force fields in which the amount of work required in going from one point to another point is not a function of the path taken. This is essentially a holomorphic field. However, it's usually a topic in the reals involving gradients, so is of little importance here. Just casting about I'm afraid.

I have read that Roger Penrose asserts a stronger position in which complex numbers and functions are somehow fundamental in reality.

But, I'm not a physicist. However, in going to 2D Euclidean space you lose multiplication of "points" as well as division.

I dabble in complex analysis all the time, but @apokrisis knows much more about the physics of this topic than I do.

When was the last time you saw an honest post in the philosophy of logic sub-category? — Shawn

Many if not most arguments here are either straight-forward or fallacious (proving one's hypothesis, etc.)

Be careful what you wish for. If you conjure up @TonesinDeepFreeze be prepared for a serious technical discussion.

I imagine that routinely goes on at the NSA. — Shawn

Not that I know of. But I'm sure it's coming. :worry:

Jack, I meant my post to be humor - the line by Justice Potter Stewart in 1964 reverberates through time. For me, to try to pick apart acts of creativity is like efforts to create computer programs to produce new mathematics. Perhaps AI will indeed subjugate us all in the future.

I hope you find a suitable place to live. I know that's got to be a big concern.

Creativity is like pornography - you'll know it when you see it. :cool:

. Many who hold deterministic views propose a "block universe" where past, present, and future exist simultaneously, eternal and unchanging. But if this is so, how does an edge of a block, explain the edge of the other side? — Richard B

Here we can see a fallacy in popularized science. To make a concept more or less understandable to the general public such images are employed. "Curved space" is another misleading notion - we all are familiar with the Earth sitting in a basketball net. Curved spacetime is legitimate, however. The injudicious use of the word "block" conjures up peculiar questions.

It's a conundrum: how to convey sophisticated scientific concepts so the public will have a "feel" for them?

Initially to see what mathematics ideas crop up in a philosophy crowd.

That's been entertaining.

The cup has one handle IFF _____? — Agent Smith

That's an easy one: IFF it is topologically equivalent to a doughnut.

What is the probability some aliens are here now AND they are homeless?

P=1

(ask Gov. Abbott)

You mention "art and creativity" almost interchangeably when art is, in fact, only one type of creativity. Science, history, play, etc are also types of creativity — 180 Proof

:ok: :cool:

When you "decided to extend this idea to a more general realm" (specific-to-general) you were doing Inductive Reasoning, — Gnomon

I don't think inductive reasoning in its usual interpretation is appropriate in this comment. I am not arguing that since it works here and there in a number of different venues it probably works in general. I am going the other direction and investigating, using deductive logic, whether ideas hold in various more or less distinct areas.

PS__"Philosophical Chatter", as you put it, seems to be how philosophers get involved in discovering new ways to look at old ideas. Are the mathematician's chalk-marks on the blackboard more involved than text-marks on a philosophical forum? — Gnomon

Good point. Nice and appropriate comparison. The chatter I was referring to was efforts to abstract "creativity" in math without using actual examples.

If it a feeling, like joy and pleasure, or something else? — Mikie

Ineffable. See the appropriate thread for an explanation.

The assumption of numbers is just a useful fiction employed by mathemagicians, which allows the ontology of Platonism to overrun the sciences. — Metaphysician Undercover

Like the Four Horsemen, leaving a trail of despair . . . :cry:

We become aware of individual objects, and infer their quantitative relationship to a collection of objects. Then we can deal with the group as-if it was a singular object (set ; whole system ; holism). So, maybe once we discover the "basic idea" of objective things & groups, we can discover (create) their subjective value (meaning) to the observer — Gnomon

Sorry, good try, but an appreciation of creativity and discovery comes with involvement, not philosophical chatter.

When we say "2+2=4" we are talking about objective reality, — Art48

Oh oh. Metaphysician Undercover might challenge that! :gasp:

At 85 not feeling bad means happy :cool:

I don't speak for formal logic. However, original research is normally required for a doctorate in math. The word "original" implies at least a small amount of creativity, as well as discovery.

Creativity: The Essence of Mathematics

Understanding Creativity in Mathematics

An example of minor creativity

Maths and formal logic are exemplars of disciplines that don't afford much importance to creativity. They are very A to B — Benj96

This thinking is so shallow it sinks below wrong. :roll:

Numbers and other mathematical entities are not a thing we talk about but a way of talking — Banno

Depends on the definition of "thing". My thing may not be your thing, but anything is better than nothing. :cool:

Was mathematics invented or discovered? :
Both discovered and invented — Gnomon

More or less, although most math people give this question little thought. In my case, I was introduced to a notion years ago in my PhD studies. A little later on I decided to extend this idea to a more general realm - a sort of creative step. Once the basic ideas of the concept were set, then came the acts of discovery - finding what flows forth logically.

The varied origins of mathematical concepts involve both complicated regressions and sheer moments of genius. My advisor would say there are no original ideas in the subject - they all trace back to antiquity (I would disagree). Take the way we measure distances (Euclidean); some time back the notion of an abstract "distance" was conceived, in a space of "objects". Thus was born the study of metric spaces, which eventually led to or coincided with the development of topological spaces. And on and on . . .

Either the chain is infinite or there was a first cause. — NOS4A2

In mathematics an infinite regression can have a "first cause".

${{F}_{n}}(z)={{f}_{1}}\circ {{f}_{2}}\circ \cdots \circ {{f}_{n}}(z)$, $n\to \infty$

$z$ is a "first cause". For any value of n one starts with z and employs backward recursion to mimic nature and arrive at a current condition.

↪Gnomon
Have you looked at ↪Art48
's book? What do you make of it? — Banno

What book?

I dread the day when mathematicians no longer produce and prove theorems, but rather attempt to interpret theorems and concepts generated by AI, some of which may be beyond human capabilities. The infamous Four Color Theorem was a first step.

When we drill down to the deepest level of matter, we get the quantum wavefunction, a mathematical object that uses regular numbers (“real” numbers) and “complex” numbers which are based on “i”, the square root of negative one. We begin with matter, trace its source, and end up with a complex-valued mathematical wavefunction — Art48

Which is not matter, just a way to predict changes in matter, energy, etc.

I keep bringing up my late Corgi, Jake, and his inability to understand algebra, though I'm sure he tried. Humans may never "understand" matter and energy at some ultimate level, but, like Jake, they can enjoy chasing metaphysical sticks and learn how to manipulate physical entities. (He learned how to manipulate me)

jgill, looks to me like the answer is just under twelve hundred post until Wittgenstein's silence wins. — Banno

He's always had my admiration for his military heroics. Now his "silence" speaks volumes.

I once had a Corgi named Jake. Although I tried teaching Jake elementary algebra, I don't think he ever got it. Nevertheless, he led an exemplary life and amused us all with his doggy wit.

I remain a tad mystified that issues straight out of medieval scholasticism keep cropping up. Especially when serious and worthwhile efforts are being devoted to untangle the conundrum of "ineffable". :chin:

In mathematics, an infinite regression can be thought of as an infinite composition of the form
$\underset{k=1}{\overset{n}{\mathop{\Re }}}\,{{f}_{k}}(z)={{f}_{1}}\circ {{f}_{2}}\circ \cdots \circ {{f}_{n}}(z)$ in which evaluation is by backward recursion and n can be as large as one wishes.

This is a causal chain that may ultimately have little to do with the "original" value or values of z occuring in the distant past. So, a specific first cause may be unnecessary for a current situation.

Theology botches it all up, IMO.

. . . . climatologists have thoughts on these arguments, and spend 30 minutes on their words — Mikie

Probably better not to use this archaic word. When I studied climatology at the U of Chi eons ago it was the post-graduate basket-weaving course in meteorology, in contrast to atmospheric science and fluid dynamics which were much more rigorous. These days more legitimate scientists are involved in climate science.

Some oldtimers (conservative Repubs?) still think of "climatology" the way it used to be - not very good at specifics and mostly just descriptive.

We’re not panicked enough, in fact — Mikie

Just what the world needs: more panic. That'll do it.

Each person is at each moment capable of . . . perceiving everything that is happening everywhere in the universe

A bit of a stretch wouldn't you say? Even what is happening in one's own body is largely below the thresholds of consciousness.

However, for some people, such experiences may lead to a sense of there being other levels of reality beyond the physical perceived in day to day consciousness. — Jack Cummins

True enough.

Nice Works of Art. But I have no idea what you are talking about. :roll:

I doubt many think of schizophrenia unless it appears in their lives and becomes something with which they must cope, either personally or by acquaintance. To have to deal with the condition is not pleasant and might make one antischizophrenic , or not. It's not an issue that impacts much of society - like racism does - and so the expression "dominant ethos" could be misleading.

Of course, being creative in any subject is another matter and is more akin to the arts and cannot be reliably taught. — Janus

An inquiring mind is a springboard to creativity.

I have no interest in the trivial question of effability vs ineffability. — Janus

Don't give up the ship, mate !

At 1,000 posts all will be revealed. The difference between effable and ineffable is ineffable, or is it effable? I can't wait !

Just grounding the thought experiment in some real numbers — Andrew M

Thanks for your input. It's a subject I haven't studied beyond the basics of time dilation.

Or are you telling me that after 30 pages you guys still don't have it sorted out? — busycuttingcrap

Don't look at me. I tried to discourage the reams of babble that emerged early on, to no avail.

↪Banno
I'll bet you are! As you should be- 30 pages on the ineffable is a solid showing! — busycuttingcrap

But a solid showing of what ?

It is an irrational sequence (never repeating itself exactly - due to the influence of emergent phenomena, but constructed by rational numbers - finite/real objects) — Benj96

Sorry, but this makes no sense. Irrationality sequence

My chickens are conscious, but they don't say much. — Banno

Thank heaven for that! However, clucks are a kind of primitive language amongst domesticated fowl. :cool: