A circle is a very close approximation of Pi which is infinity itself. — invicta

This is gobbledygook. But I would not be surprised were you unable to see that.

Gnomon
A circle is a very close approximation of Pi which is infinity itself. — invicta
This is gobbledygook. But I would not be surprised were you unable to see that. — Banno

As usual, we have here a vocabulary conflict between people with opposing points of view. Such disagreements are not resolved by disparagement. The word "infinity" has several definitions, depending on context*1.

I suspect that is using the term "Infinity" to mean simply a never-ending series of numbers*2. Perhaps a better word would be "indefinite". His colloquial usage is different from your technical version, but not meaningless "gobbledygook"*3. My tongue-in-cheek "Pi is infinite" link may be somewhat "gobbledy", but it expressed a notion that may be closer to Invicta's usage regarding "infinite regress". The number PI is a never-ending series, but in practice it has a finite value*4. PI as a concept is irrational, and in a general sense unbounded or infinite*5. :wink:



*1. What is Infinity?
Infinity is an idea of something that has no end. In general, it is something without any bound. It is a state of endlessness or having no limits in terms of time, space, or other quantity.

In Mathematics, “infinity” is the concept describing something which is larger than the natural number.

*2. Infinity : a number greater than any assignable quantity or countable number (symbol ∞).

*3. Gobbledygook : language that is meaningless or is made unintelligible by excessive use of abstruse technical terms; nonsense.

*4. PI infinite :
How is pi infinite? It’s not infinite in value. It’s more than 3 and less than 4, so its numerical value is certainly finite. What’s infinite about it is the amount of time, or more precisely the amount of calculation, that it would take to express its exact value.
https://www.quora.com/How-is-%CF%80-infinite-if-a-circle-has-finite-area

*5. PI is irrational :
Hence, pi is a real number, but since it is irrational, its decimal representation is endless, so we call it infinite.
https://www.scienceabc.com/pure-sciences/do-we-have-any-mathematical-proof-that-pi-is-infinite.html

π is a number (real), ∞ is not, hence they're not the same.
√2 and e are other examples of irrational real numbers, not ∞ either.
⅓ or 1/3 is a rational number that can't be expressed by a finite string in ordinary decimal notation.

If we suppose there are no circles (in nature), then there are some other shapes instead.
What shapes are possible (in nature) anyway? Are straight lines?

The concept of self-organization might interest you. In this book, a theoretical physicist looks into the cosmology of Jacob Boehme. He makes the argument that Jacob Boehme's conception of a self-organizing world might end up being more important in the long run than his contemporary Galileo.

Or for something a bit more down to earth there is Erich Jantsch's "The Self Organizing Universe," which I have heard good things about, although it is a bit dated.

Melanie Mitchell's Complexity: A Guided Tour is very good to, but isn't trying to look at any sort of big picture. But, it provides a look at how "a circle can draw itself," in a systems perspective.

In Mathematics, “infinity” is the concept describing something which is larger than the natural number. — Gnomon

Huh?

A circle is a very close approximation of Pi which is infinity itself. — invicta

You would deny Invicta the privilege of meaning what he says?

Then you are harder on him than even I.

The tools of philosophy are the words they use, so it is best we use them with due diligence.

The account in the OP is as follows:

...if infinite causes are the chain of sequences ad infinitum does such a chain not imply a closed loop... — invicta

And the clear answer is no, it doesn't. There is a difference between an infinite progression and a loop.
When this was pointed out, Invicta doubled down:

If you only knew Pi, which you obviously can’t as it’s irrational and infinite …could you draw a circle? — invicta

Of course, we do know pi. A formula for it was given above, and the definition is the subject of primary school mathematics. The discussion continued with Invicta playing on the two meanings of "irrational", only to arrive at

A circle is a very close approximation of Pi which is infinity itself. — invicta

...which as i said, is gobbledegook. A circle is no more an approximation to pi than a fish is an approximation to a democracy; And Pi is not infinity itself.

Philosophy is about getting the words right. you, @Invicta and have yet to understand this.
And isn't far ahead of you.

Philosophy is about getting the words right — Banno

That is not the sum total of the subject although it's an important part.

Aristotle's argument for a first cause is based on the observation that motion and change are fundamental features of the natural world and are characteristic of everything that we observe. He believed that motion and change cannot occur without some cause or explanation, and that this cause must itself be unmoved and unchanging. Aristotle argued that if there were no such first cause, then motion and change would be infinite and eternal, without any ultimate explanation or source.

I think part of the reason for his insistence that the first mover must itself be unmoving is derived from the Phaedo. In that, Socrates suggests that in order to explain a particular phenomenon, one must have knowledge of a more general principle or cause that underlies it. Socrates refers to this more general principle as the "cause" or "explanans," and the particular phenomenon as the "effect" or "explanandum."

Socrates asserts that the explanans must be of a higher order than the explanandum, because it is the more general principle that explains why the particular phenomenon occurs. By this logic, if the unmoved mover itself was subject to motion and change, then it would provide no explanation for these, as it would itself be part of what we are required to explain.

So in answer to the question, I don't think Aristotle's principle of the first cause can be equated with the uroboros, and the idea of 'self-causation' in respect to such an analogy is muddled.

Cheers. I'll take your word for the exegesis, and move on, since Aristotle's notions of causation are somewhat anachronistic. That is, our understanding of causation has moved on somewhat. My involvement here is to point point out the obvious poverty of some of the arguments.

That is not the sum total of the subject although it's an important part. — Wayfarer

Yep.

In Mathematics, “infinity” is the concept describing something which is larger than the natural number. — Gnomon
Huh? — jgill

Sorry, I forgot to provide a link. That quote came from a math website. I didn't make it up. :smile:

Infinity is not a real number, it is an idea. An idea of something without an end. . . .
{1, 2, 3, ...} The sequence of natural numbers never ends, and is infinite.
https://www.mathsisfun.com/numbers/infinity.html

A circle is a very close approximation of Pi which is infinity itself. — invicta
You would deny Invicta the privilege of meaning what he says? — Banno

No. As I pointed out above, the meaning of "infinite" varies depending on context and intent. So, I'm merely allowing to use the word in a way that suits his context. I don't necessarily agree with his conclusions, but I want to hear his argument -- in his own words -- not necessarily in my personal vocabulary. :smile:


Infinity is not a real number, it is an idea. An idea of something without an end. . . .
{1, 2, 3, ...} The sequence of natural numbers never ends, and is infinite.
https://www.mathsisfun.com/numbers/infinity.html

Disagreements about terminology are unnecessary. Discussants can instead acknowledge the clear definitions in mathematics:

INFINITE:

Df. x is finite if and only if x is 1-1 with a natural number

Df. x is infinite if and only x is not finite

Df. x is countable if and only if (x is finite or x is 1-1 with the set of natural numbers)

Df. x is denumerable if and only if (x is infinite and x is countable)

Df. +inf is an element in the extended real system such that for any real number x we have x < +inf. Note that for any specification of 'the extended real system', +inf can be any object other than a real number or -inf (see below), and +inf does not have to be infinite (in the sense of 'is infinite' defined above).

Df. -inf is an element in the extended real system such that for any real number x we have x > -inf. Note that for any specification of the 'the extended real system', -inf can be any object other than a real number or +inf (see above), and -inf does not have to be an "infinitesimal" (and 'infinitesimal' is not defined anyway, except in non-standard analysis, which is different from the extended real system).

CIRCLE:

Df. L is a circle if and only there exists a plane and member x of that plane such that L is the set of points all equidistant from x

Df. if L is a circle, then x = G(L) ('the origin of L') if and only if all members of L are equidistant from x

Df. if L is a circle, then x = C(L) ('the circumference of L') if and only if there is an increasing sequence of lengths of perimeters of inscribed polygons such that x is the limit of the sequence

Df. If L is a circle, then x = D(L) ('the diameter of L') if and only if x a line through G(L) such that x ends on both ends on members of L

PI:

Df. x is a real number if and only if x is an equivalence class of Cauchy sequences of rational numbers

Th: if x is a real number, then x is infinite

Note: The above theorem is an artifact of the construction of the reals as equivalence classes. The theorem only says that the CARDINALITY of x is infinite; it does NOT say that any real number has infinite MAGNITUDE. Cardinality and magnitude are DIFFERENT. No real number has infinite magnitude.

Df. if x is a member of the extended reals, then x has finite magnitude if and only x is a real number

Df. if x is a member of the extended reals, then x has infinite magnitude if and only if (x = +inf or x = -inf)

Th. if x is a member of the extended reals, then x has finite magnitude if and only if x is a real number

Th. if x is a member of the extended reals, then x has infinite magnitude if and only if (x = +inf or x = -inf)

Df. x = pi if and only if for any circle L, x = C(L)/length(D(L))

Th: pi is infinite

Note: The above theorem is an artifact of the construction of the reals as equivalence classes. The theorem only says that the CARDINALITY of pi is infinite; it does NOT say that pi has infinite MAGNITUDE. Cardinality and magnitude are DIFFERENT. pi does not have infinite magnitude.

Th: pi has finite magnitude

Df. if x is a real number, then s = the decimal expansion of x if and only if [fill in the definiens here of the well known notion of a certain denumerable sequence]

Th: The decimal expansion of pi has no terminal repeating subsequence

Note: The above theorem is an artifact of the construction of the reals as equivalence classes. The theorem only says that the CARDINALITY of pi is infinite; it does NOT say that pi has infinite MAGNITUDE. Cardinality and magnitude are DIFFERENT. pi does not have infinite magnitude. — TonesInDeepFreeze

Nice, Tones. I'll go along with that. Thanks.

I added this:

Df. if x is a member of the extended reals, then x has finite magnitude if and only x is a real number

Df. if x is a member of the extended reals, then x has infinite magnitude if and only if (x = +inf or x = -inf)

Th: pi has finite magnitude

Disagreements about terminology are unnecessary. Discussants can instead acknowledge the clear definitions in mathematics: — TonesInDeepFreeze

Math definitions will not resolve the terminology disputes in this thread because is not making a mathematical proposition. "Infinite Regress" and "First Cause" are philosophical concepts that are not addressed by Mathematics : the abstract science of number, quantity, and space. That may be why such open-ended (infinite???) concepts are annoying to some posters, since it can't be ruled True or False by numerical authority. Satisfactory (not true or false) answers will depend as much on intuition as on logic.

As evidenced by never-ending dialogues on this forum, some philosophical questions, including definitions, are often as clear as mud. If metaphysical meanings were as "clear" as math, we wouldn't have forums for extended argumentation. Instead, we could just "shut up and calculate". For those who prefer the clarity of Math, here's a nice online discussion forum : :smile:

Wolfram Community :
https://community.wolfram.com/content?curTag=mathematics

The Metaphysics of Stephen Wolfram :
https://www.youtube.com/watch?v=-jNMh8uuqQY

"Pi is infinity" is not a mathematical proposition?

Or will you claim anything in order to defend your account?

The gravamens may not be mathematical, but certain mathematical points have entered in. As to the mathematical points themselves, agreed upon definitions are crucial. If some of the same terminology is used in philosophical senses different from the mathematical senses, then each time it should be made explicit which sense is active.

Pi is mathematical. And 'is infinite' has both a mathematical sense and philosophical senses. Whatever else one may wish to say philosophically about Pi and the notion of infinity it is at least a good starting place to reference the mathematical definitions. And by doing that, I cleared up a common confusions about mathematics:

(1) There is a distinction between the adjective 'infinite' and the noun 'infinity'.

For the adjective, we define:

S is infinite if and only if S is not finite.

For the noun, we define by choosing objects to serve as +inf and -inf in the extended reals.

The adjective and the noun should not be conflated. In particular, other than the extended reals, in ordinary mathematics, there is no object named "infinity". An such infinite sets such as the set of natural numbers is not named "infinity".

(2) There is a distinction between cardinality and magnitude.

Pi has finite magnitude.

Pi has an equivalence class of Cauchy sequences of rationals is infinite (has infinite cardinality).

(3) The decimal expansion that represents Pi (as Pi is the limit of the sequence) is infinite and has no finite initial subsequence that converges to Pi.

One can say whatever philosophy about the notion of infinity, but when it's mixed with mathematics, such as Pi, Circles, sequences, etc., the one should at least get the mathematics right before invoke it in one's philosophizing.

Pi is not a circle
— Banno

Of course it’s a circle — invicta

"Pi is a circle" is a mathematical claim, not a philosophical claim.

Mathematical definitions of 'Pi' and 'is a circle' are crucial for settling a dispute about the claim "Pi is a circle".

↪Gnomon
"Pi is infinity" is not a mathematical proposition?
Or will you claim anything in order to defend your account? — Banno

As previously noted, I interpret his use of "infinity" as a philosophical postulation, not a mathematical proposition. Apparently, your more restrictive*1 vocabulary (your account) does not allow that distinction. :smile:


*1. Two-Valued Logic is Not Sufficient to Model Human Reasoning
https://ceur-ws.org/Vol-1651/12340059.pdf

If you would like to make some point about what is written in a paper, then you can say what that point is, rather than just have me scurry and sidetrack to read something that you haven't said what about it you think is important to consider relative to what I've said.

, I interpret his use of "infinity" as a philosophical postulation, not a mathematical proposition. — Gnomon

What could that mean? Pseudo-scientific garbage? New age postulating?

Certainly that's not philosophy, in anything more than a "pop" sense; though it is apparent from your odd threads and faux footnotes¹ that you do not understand.

Philosophy is not "making shit up", as so many here seem to think.

1. What was the remainder of the name of that article, the bit you intentionally left out? What was the topic of that article? What point did you think you were making by the pretentious footnote?

Go back again; in the OP posits that a linear progression of causation is the same as a closed loop of causation. That is wrong.

He then equate pi with circles and with infinity. Both of these are at best misleading.

Philosophy is not about misusing terms, as seems to think.

Math definitions will not resolve the terminology disputes in this thread because ↪invicta is not making a mathematical proposition. "Infinite Regress" and "First Cause" are philosophical concepts that are not addressed by Mathematics — Gnomon

Wrong. I and others have studied infinite regress in detail, as infinite compositions or iterations.

In Mathematics, “infinity” is the concept describing something which is larger than the natural number. — Gnomon
Huh? — jgill

Sorry, I forgot to provide a link. That quote came from a math website. I didn't make it up. — Gnomon

What is "the natural number" ?

Mathematics has developed a fair understanding of ∞. But ∞ can be a few different things. Cantor (ℵ cardinals) showed that there are infinite different infinites, no less. So, in a concise context, ∞ is ambiguous. The rules of the natural numbers, N (or the real numbers, R) don't apply to ∞s. That's what mathematics taught us.

Dedekind's definition of infinite:

• a set is infinite if and only if there is a bijection between the set and a proper subset of the set
• |S| = ∞ ⇔ ∃ ƒ (bijection): S → T ⊂ S

The even numbers is a proper subset of the naturals, and there's a one-to-one mapping between them, hence the naturals is infinite. Might also be what you wrote, . Also related to equinumerosity.

Tarski came up with another concise definition that can be shown identical to Dedekind's.

I didn't mention Dedekind infinitude, only because I had too much to cover already.

x is infinite <-> x is 1-1 with a natural number. [call this 'Tarski infinite']

x is Dedekind infinite <-> x is 1-1 with a proper subset of x

I don't like the use of the lemniscate as you do, because it invites conflating the point of infinity in the extended reals with a cardinal. Also, we don't write 'card(S) = leminscate' to say that S is infinite. As you mention, there are infinite sets of different cardinalities, so it can't be the case that there is just one object (named by the lemniscate) that all infinite cardinalities are equal to.

Tarski came up with another concise definition that can be shown identical to Dedekind's. — jorndoe

Tarski's definition and Dedekind's definition are not equivalent in ZF but they are equivalent in ZFC.

, I was thinking of this Tarskian definition, defining finite, then infinite from that...

S is a set

℘(S) is the set of all subsets of S including ∅ and S itself
- the power set, Weierstraß, Cantor

F ⊆ ℘(S) is a family of subsets of S

m ∈ F is a minimal element of F ⇔ ∀ x ∈ F [ x ⊄ m ]
- no smaller subset

M(F) = { m ∈ F | x ∈ F ⇒ x ⊄ m }
- the set of minimal elements

S is finite ⇔ ∀ F ⊆ ℘(S) [ F ≠ ∅ ⇒ M(F) ≠ ∅ ]
- a set is finite if and only if every non-empty family of its subsets has a minimal element, Tarski

S is infinite ⇔ S is not finite

I'm guessing that's what you had in mind also. (?)

Anyway, didn't mean to distract the regress, ouroboros, etc discussion.

If you would like to make some point about what is written in a paper, then you can say what that point is, rather than just have me scurry and sidetrack to read something that you haven't said what about it you think is important to consider relative to what I've said. — TonesInDeepFreeze

All I had to say is in the title. :smile:

*1. Two-Valued Logic is Not Sufficient to Model Human Reasoning
https://ceur-ws.org/Vol-1651/12340059.pdf

Wrong. I and others have studied infinite regress in detail, as infinite compositions or iterations. — jgill

Irrelevant ! I didn't intend to defend 's conclusions, but just to defend his right to use a colloquial meaning of "infinite" in a philosophical proposition, without being challenged to present a mathematical or scientific justification. The OP presents metaphors of snake-circles, not mathematical proofs. Is "self-caused" a mathematical concept?

Obviously, what has incensed some posters in this thread is the supernatural implications of the OP. Which they hope to demolish by turning a broad philosophical question into a narrow technical definition. Even though the universe is now known to have a finite beginning in spacetime, some thinkers like to think of it as infinite/eternal, so they don't have to deal with open-ended questions such as the OP. :smile:

PS___I usually find you to be more open-minded than the True/False debunkers. In a different thread, one poster asserted that "the opposite of science is pseudoscience". Which is indicative of either/or ; two-value reasoning. In that case, Philosophy must either present empirical evidence, or be rejected as Pseudoscience. Personally, I view Philosophy as complementary to Science, using different methods.

Colloquial : (of language) used in ordinary or familiar conversation; not formal or literary. ___Oxford

STUDENTS’ COLLOQUIAL AND MATHEMATICAL DISCOURSES ON INFINITY AND LIMIT
https://files.eric.ed.gov/fulltext/ED496910.pdf

What is "the natural number" ? — jgill

I'm not a mathematician, so I don't discriminate between Natural and Super-natural numbers. Is Infinity literally supernatural? :smile:

In mathematics, the natural numbers are the numbers 1, 2, 3, etc., possibly including 0 as well. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to ... ___Wikipedia

Usually when someone posts a link, it is taken as a suggestion to visit that link. So, without you saying what specifically you wanted me to take from the article, I would have detoured to read and study an article of which all you mean to say is what you posted anyway.

And nothing I've said depends on claiming that two-valued logic is sufficient for modeling human reasoning.

ght to to use a colloquial meaning of "infinite" in a philosophical proposition, without being challenged to present a mathematical justification. — Gnomon

It wasn't just philosophical. There is a mathematical context also, whether primary or secondary. Especially the assertion that Pi is a circle is mathematical.

incensed — Gnomon

I don't know who you think was incensed.

In your definition of the set of minimal elements, you mistakenly left out a quantifier over 'x'.

The formula should be:

M(F) = {m e F | Ax(x e F -> ~ x proper subset of m)}

Then the rest of the definition of 'is finite' is good.

Yes, as you might know, in a 1924 paper, Tarski stated the definition and proved its equivalence with other definitions, as discussed in Suppes's great intro text book 'Axiomatic Set Theory'.

It also turns out that Tarski's definition is equivalent to:

S is finite <-> S is 1-1 with a natural number.

Usually when someone posts a link, it is taken as a suggestion to visit that link. So, without you saying what specifically you wanted me to take from the article, I would have detoured to read and study an article of which all you mean to say is what you posted anyway. — TonesInDeepFreeze

That is not my intention for using links. Instead, I try to say what I have to say in the post, and then add the links "for further reading"--- if someone is interested in more detail. In contentious threads like this though, where my limited knowledge will be challenged & dismissed, another function of links is to let the experts speak on the same topic, with the kind of authority I lack. :smile: