You mean because we can travel around a circle forever? I mean infinite more like if absolute coordinates existed we could go straight in one direction without reaching a limit.

Circumnavigation. There is no edge ("limit") to a sphere.

Nonmathematical infinity seems to be a category mistake. Infinity is a mathematical concept, oui?

If anyone persists in this madness, s/he would need a nonquantitative definition of limit.

The obvious question: is $\infty$ a number?

Have a dekko at the following:

6 and 2 are numbers

$6 + 2 = 8$
$6 - 2 = 4$
$6 \times 2 = 12$
$6 \div 2 = 3$

Now $\infty$

$\infty + \infty = ?$
$\infty - \infty = ?$
$\infty \times \infty = ?$
$\infty \div \infty = ?$

$\infty$ isn't a number like 2, 3, 4, 6, 8, 12.

Don't get me started on $0$!

Unless infinity is formally identified with a finite piece of syntax, whereupon becoming a circularly defined and empirically meaningless tautology, infinity cannot even be said to exist inside mathematics, let alone outside.

Potential infinity, as the intuitionists keep stressing and as programmers demonstrate practically, is the only concept that is needed, both inside and outside of mathematics, that refers to finite entities of a priori indefinite size.

The myth of absolute infinity is what give the illusion of mathematics as being an a priori true activity that transcends Earthly contingencies.

* There are rigorous definitions of 'limit', in various contexts, in mathematics.

* To my knowledge, there is no general mathematical definition of 'is a number'. However 'ordinal number', 'cardinal number' and the predicate 'is infinite' have rigorous mathematical definitions, and there are proofs that there are ordinal numbers and cardinal numbers that are infinite.

* The lemniscate [here I'll use 'inf'] does not ordinarily denote a particular object. Probably its two most salient uses are for (1) points on the extended number line and (2) in expressions such as "the SUM[n = 1 to inf] 1/(2^n). With (1), inf and -inf can be any arbitrary objects (they don't even have to be infinite) that are not real numbers, serving as points for the purpose of a system. With (2), 'inf' is eliminable as it is merely convenient verbiage that can be reduced to notation in which it does not occur.

So your examples of arithmetic involving inf are not meaningful. However, there are rigorously defined operations of ordinal addition, subtraction, multiplication, and division.

* I agree that you in particular are better off not speaking on 0!, which also has a rigorous mathematical treatment.

* 'infinity' as a noun does not ordinarily have a mathematical definition, though 'is infinite' does. A mathematical definition is never circular nor a tautology.

* The notion of 'potentially infinite' is of course central to important alternatives to classical mathematics. However, as far as I know, formalization of the notion is not nearly as simple as the classical formalization of 'infinite'. Therefore, if one is concerned with truly rigorous foundations, when one asserts that the notion of 'potentially infinite' does better than that of 'infinite' one should be prepared to accept the greater complexities and offer a particular formalization without taking it on faith that such formalizations are heuristically desirable, as we keep in mind that ordinary mathematical application to science and engineering uses the simplicity of classical mathematics as one first witnesses in Calculus 1.

* What writings by intuitionists are fairly rendered as describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" of any kind?
.
* One may take mathematics as a priori true without commitment to infinite sets.

Do straight lines exist? And even if you traveled the earth forever you will see the same places more than once.

Do straight lines exist? And even if you traveled the earth forever you will see the same places more than once. — TiredThinker

Take another sip, my friend, then off to bed. :yawn:

I was simply pointing out that basic arithmetic operations are undefined for $\infty$ which, to me, implies it (infinity) isn't a number like 2, 3, 18986, 0.98457..., 1/8, etc.

Thanks.

You just went right past what I wrote.

You just went right past what I wrote. — TonesInDeepFreeze

I'm just puzzled/intrigued by the fact that you can't do math with nihilism and also with $\infty$.

I merely stated the needed corrections to your uninformed argument.

I merely stated the needed corrections to your uninformed argument — TonesInDeepFreeze

What's so uninformed about:

I'm just puzzled/intrigued by the fact that you can't do math with nihilism and also with $\infty$ . — Agent Smith

?

You're trolling. What was uninformed is the post::

Thanks for engaging with me. If I can think of anything worth your time, I'll post you a reply. Until then adios!

Buh bye.

Do straight lines exist? — TiredThinker

AFAIK, there are "straight lines" only in the abstract Euclidean space. However, the shortest path between any two points is a geodesic ...

And even if you traveled the earth forever you will see the same places more than once.

The circumferential path does not end (i.e. it's in-finite); no point ("place") on the path is a boundary, therefore the path is unbounded.

'infinity' as a noun does not ordinarily have a mathematical definition, though 'is infinite' does. A mathematical definition is never circular nor a tautology. — TonesInDeepFreeze

In which case, you surely agree that absolute infinity isn't a semantically meaningful assignment to a mathematical entity, for any semantic interpretation of the symbol of infinity as referring to extensional infinity, is question begging.

* The notion of 'potentially infinite' is of course central to important alternatives to classical mathematics. However, as far as I know, formalization of the notion is not nearly as simple as the classical formalization of 'infinite'. Therefore, if one is concerned with truly rigorous foundations, when one asserts that the notion of 'potentially infinite' does better than that of 'infinite' one should be prepared to accept the greater complexities and offer a particular formalization without taking it on faith that such formalizations are heuristically desirable, as we keep in mind that ordinary mathematical application to science and engineering uses the simplicity of classical mathematics as one first witnesses in Calculus 1. — TonesInDeepFreeze

The semantic notion of absolute infinity (whatever that is supposed to mean) isn't identifiable with the unbounded quantifiers used in classical mathematics, logic and set theory, due to the existence of non-standard models that satisfy the same axioms and equations without committing to the existence of extensionally infinite objects. Science and engineering continues to work with classical mathematics , as well as classical logic, due to their vagueness, simplicity and brevity as a junk logic for crudely expressing ideas, which usually cannot be fully formulated or solved in those notations due to the inconvenient truths of software implementation and physical reality. Most software engineers don't regard themselves to be mathematicians or logicians, due to historical reasons concerning how mathematics and logic were initially conceived and developed.

* What writings by intuitionists are fairly rendered as describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" of any kind? — TonesInDeepFreeze

SEP's article on intuitionism is a useful introduction for understanding the notion of Brouwer's tensed conception of mathematics, i.e. mathematics with lazy evaluation, that rejects both formalism and platonism, in which unbounded universal quantification is understood to refer to potential infinity, which leads to his formulation of non-classical continuity axioms. In a similar vein, Edward Nelson's Internal Set Theory adds tenses to Set Theory, by distinguishing the elements of a set that have so far been constructed that have definite properties, from those that will potentially be constructed in the future, that have indefinite properties.

A groom, hand on heart, vows sincerely to the bride " I will always remain faithful". Later that afternoon, he runs off with the bridesmaid. Did he really contradict his earlier vows, or does a contradiction exist only in the minds of those who misconceive the nature of infinity ?

Perhaps one might say that to view the groom as contradicting his earlier vows amounts to a definition of 'negated absolute infinity' - but this interpretation is unnecessarily problematic in asserting the negation of a statement that isn't a verifiable proposition with verifiable meaning.

However, the shortest path between any two points is a geodesic ... — 180 Proof

On a curved surface. Straight lines otherwise. We're talking 3D, not 4D space-time.

Science and engineering continues to work with classical mathematics , as well as classical logic, due to their vagueness, simplicity and brevity as a junk logic for crudely expressing ideas . . . — sime

I'm willing to concede that my colleagues and I have produced mathematical contributions that are worthless, but calling classical mathematics "junk logic" and "crudely expressing ideas" is a ridiculous accusation. On the other hand, that may not be what you are saying. It's hard to work through some of your lengthy paragraphs. Probably just me.

Unless infinity is formally identified with a finite piece of syntax, whereupon becoming a circularly defined and empirically meaningless tautology, infinity cannot even be said to exist inside mathematics, let alone outside — sime

In the complex plane "infinity" is called "the point at infinity" and correlates directly with the north pole of the Riemann sphere - a specific point. But I've never used this concept.

I'm willing to concede that my colleagues and I have produced mathematical contributions that are worthless, but calling classical mathematics "junk logic" and "crudely expressing ideas" is a ridiculous accusation. On the other hand, that may not be what you are saying. It's hard to work through some of your lengthy paragraphs. Probably just me. — jgill

sure, its an overstatement born of frustration with somewhat outdated formal traditions that still remain dominant in the education system.

I am not interested in an infinite trip. I am asking about and infinite landscape. Only new information all the time.

I am asking about and infinite landscape. Only new information all the time. — TiredThinker

:chin: Such as unknown unknowns which necessarily encompass "knowns":

The more we learn about the world, and the deeper our learning, the more conscious, specific, and articulate will be our knowledge of what we do not know; our knowledge of our ignorance. For this indeed, is the main source of our ignorance - the fact that our knowledge can be only finite, while our ignorance must necessarily be infinite. — Karl Popper

Are uncountable infinities mathematical?

* What writings by intuitionists are fairly rendered as describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" of any kind?
— TonesInDeepFreeze

SEP's article on intuitionism is a useful introduction for understanding the notion of Brouwer's tensed conception of mathematics — sime

That article does not describe potentially infinite sets or sequences in a manner that is properly paraphrased as "finite entities of a priori indefinite size" or as "finite entities".

Moreover, most remarkably, in another thread:

https://thephilosophyforum.com/discussion/comment/653236

you say, "whose length is eventually finitely bounded". Again, there is nothing in the SEP article that states anything like that.

/

absolute infinity isn't a semantically meaningful assignment to a mathematical entity — sime

Who specifically do you have in mind as mentioning "absolute infinity" since Cantor's work was superseded by formal set theory?

Moreover, my earlier remark stands and should not be overlooked:

* 'infinity' as a noun does not ordinarily have a mathematical definition, though 'is infinite' does. A mathematical definition is never circular nor a tautology. — TonesInDeepFreeze

You simply skipped the point that mathematical definitions are never circular or tautologies.

The semantic notion of absolute infinity (whatever that is supposed to mean) — sime

Whatever what is supposed to mean? "The semantic notion of absolute infinity" or "absolute infinity"? Anyway, in either case, who specifically do you have in mind as mentioning "absolute infinity" since Cantor's work was superseded by formal set theory?

existence of non-standard models that satisfy the same axioms and equations without committing to the existence of extensionally infinite objects. — sime

What specific non-standard models and specific "axioms and equations" do you have in mind?

What is your rigorous mathematical (or even non-rigorous philosophical) definition of "extensionally infinite"?

Science and engineering continues to work with classical mathematics , as well as classical logic, due to their vagueness, simplicity and brevity — sime

Formal classical mathematics is exact in the sense that there is an algorithm to check whether a given finite sequence of formulas is or is not a proof. Any alternative to classical mathematics that also has that attribute would need to be also evaluated for simplicity of the formulation of the system.

outdated formal traditions that still remain dominant in the education system. — sime

Infinite sets come into play in Calculus 1. What pedagogy would you propose for people to find derivatives without infinite sets?

Infinite sets come into play in Calculus 1. What pedagogy would you propose for people to find derivatives without infinite sets? — TonesInDeepFreeze

You mean without absolutely infinite sets, presumably. The overall approach would be to stress that the mathematical notion of a derivative approximates the real-life practice of directly measuring the quotient of two arbitrarily small intervals Dy and Dx with respect to some observed function. This is opposite to the conventional way of thinking, which construes the practice of measuring a slope as a means of approximating an ideal, abstract and causally inert mathematical derivative.

With this in mind, the classical definition of df/dx with respect to the (ε, δ)-definition of a limit, can be practically interpreted by interpreting ε to be a potential infinitesimal, and δ as representing a random position on the x axis given the value of ε , which when applied to the function yield df and dx as potential infinitesimals, i.e. finite rational numbers, whose smallness is a priori unbounded.

What about fractals? These are infinite spaces that you can explore on a computer. Their instantiation in a computer realizes what was an abstract infinity.

. . . directly measuring the quotient of two arbitrarily small intervals Dy and Dx with respect to some observed function — sime

dx may be arbitrary, but dy depends on dx and is not arbitrary. That's where the observed function comes into play. Maybe you are talking about something else. You seem confused about these things. Or maybe I misinterpret.

the classical definition of df/dx with respect to the (ε, δ)-definition of a limit, can be practically interpreted by interpreting ε to be a potential infinitesimal, and δ as representing a random position on the x axis given the value of ε , which when applied to the function yield df and dx as potential infinitesimals, i.e. finite rational numbers, whose smallness is a priori unbounded. — sime

Non-standard analysis is what you are referencing here in a somewhat befuddled fashion. What is a potential infinitesimal?