I think it's important (whether it's new to you or you've forgotten) to familiarize yourself with the Frege-Russell definition of number. Here's a link to an abstract of an article by R. L. Goodstein that provides a nice description. Note that he uses (1, 1) to stand for the more common "one-to-one".

https://www.jstor.org/stable/3609188#:~:text=Russell%20definition%20of%20number%20is%20the%20identification%20of,1%29%20related%20to%20some%20class%20of%20n%20members.

Re the OP :

For instance, the number “2” exists outside spacetime. — Art48

Others have commented on this already, but here's my take : most materialists and idealists (recognizing that there are various flavors of each) must disagree with this premise, since both camps consider mind-stuff to be part of the world (i.e., "spacetime"). Only dualists would agree that thoughts are divorced from the "world out there".

With apologies, the better question is, "Do numbers exist independent of minds?" * And of course, debate over mathematical Platonism has raged for many decades, if not centuries, and will not be answered on TPF any time soon. Still, it's fun to plant your flag and defend it. My own view, following from the Frege-Russell definition, is that numbers do NOT exist independent of minds because the act of placing objects in a one-to-one relationship is a cognitive one.

*Assuming most idealists assert that ALL is mind, their answer to the question of mind-independence must be no, correct?

I am reminded of Existence Theorems in math, where it is proved that some mathematical object, number or property must exist, but it is either not possible or unnecessary to produce said object.

... and including this cat person — Bartricks

Hahaha. Almost missed this one.

This cat person has feelings, you know. Your callous dismissal wounds me. Cannot cat people bleed?

From now on, I would prefer if you referred to me as Mr. Real Gone Cat. We are not friends, sir. Good day.

By the way, and roughly speaking, I think implication in logic is something that happens within statements (if then), whereas entailment happens within arguments, that is, between sets of statements and a conclusion. — Jamal

Pretty much in agreement, but I think you're splitting hairs a bit finely here. The statement $A\Rightarrow B$ carries within it the modus ponens argument. [Isn't modus ponens just one row of its truth table?]

Unless you're reserving "implies" for the form of the argument, and "entails" for an instantiation. (Still too delicate for my tastes.)

Yup. Been making the same point, difference being I ... <hangs head in shame> ... never read the entire OP, so never clicked the link.

So, tell me,when someone says 'imply' do you think they mean 'entail'? — Bartricks

I don't think I've ever used the word "entail" in my life.

Have you? [Hint : review this topic before you answer]

I'm sure there's a Logic for Dummies out there. Please find it.

Speaking for myself, I've made it clear that there are two uses, or meanings, for "implies". One meaning is "suggests" (Those dark clouds imply rain.). The second meaning is "necessitates". The second meaning is a stronger relationship. This second meaning is also known as Logical Implication

Since we use logic to construct our arguments (or should), logical implication is always meant when using "implies" in an argument. Otherwise, modus ponens would have been abandoned millenia ago.

Now, if you so desire, you can reserve "implies" for the looser definition (suggests) and employ "entails" for the stronger definition (logical implication), but none of your listeners will know you are doing this until you tell them.

The problem for the OP has to do with another word. The OP confuses the reader by using "cause" as a noun and "cause" as a verb without distinguishing between the two. As a noun, it is true that there are necessary causes and sufficient causes. But as a verb, "cause" indicates logical implication, i.e., the antecedent is sufficient for the consequent.

Yeah, this also happens every time a non-math person starts a math thread on TPF. As much as you try to clarify the mathematical definitions for them, they keep insisting on their own folk-wisdom interpretations. So you go round and round and get nowhere.

Ah, the oldest cause for argument in philosophy : failure to agree on a definition of terms. [What follows should be obvious to most but was seemingly forgotten in this case.]

Bartricks is using the everyday, unwashed-man-in-the-street definition of "implies". From Oxford Languages ,

imply : strongly suggest the truth or existence of (something not expressly stated):
"the salesmen who uses jargon to imply his superior knowledge" · "the report implies that two million jobs might be lost"

Invizzy is using the definition of "implies" from logic, also called the conditional. It is a much stronger relationship. In logic, if A implies B, then the truth of A guarantees the truth of B.

Since philosophy tends to favor logic, I think the stricter definition must win out, unless specified otherwise.

It's a category error because you're judging mathematical notions of infinity by some dubious metaphysical standard. One that is vague at best.

You keep coming back to a line as being "constrained" in one dimension but not another. Are you aware that a plane consists of an uncountably infinite set of lines? And 3D space consists of an uncountably infinite set of planes? Now, by your understanding, is 3D space "constrained"?

Finally, it can be shown that the cardinality of the set of points in 3D space is equal to the cardinality of points in a line. I.e., the line can be mapped onto 3D space (and vice versa). So how is the line constrained again?

Before accusing another of nonsense, try picking up a math book.

G-d? Brahman? Pleroma? This isn't woo? Merriam-Webster :

woo-woo : dubiously or outlandishly mystical, supernatural, or unscientific

But whether you value such things is beside the point. I stand by my assertion : it's a category error.

Oh, you've been comparing math to woo all along. Seems like a category error to me. Carry on.

No, I don't think this is correct.

To show that a set has the same cardinality as the set of natural numbers, you only have to show that the elements of the set can be placed in one-to-one correspondence with the set of natural numbers. You don't actually have to complete the set. To show that an infinite set has a greater cardinality, you only have to show that at least one element exists that cannot be placed in the one-to-one correspondence (ala Cantor's diagonal proof). No actual infinities need be assumed.

Friend, as litewave pointed out, by your own argument, when you name a thing, you are placing a boundary on it. If you can, name two metaphysical identities. Now count them. Two.

Anyway, too many folks are trying to explain to you why you're wrong, but you keep doubling down. It's starting to feel like piling on. I'll keep reading comments, but I think I'm out.

Why can't you count metaphysical infinity? I assume you only recognize one metaphysical infinity, so haven't you counted it? One.

I was just about to post this very same response.

Each line consists of an uncountable infinity of points, each plane consists of an uncountable infinity of lines, and 3D space consists of an uncountable infinity of planes. But two points are always discrete, two lines are always discrete, and two planes are always discrete (except where they may intersect).

A metaphysical infinity has absolutely no limits or boundaries. Due to this, it cannot be discerned as a unit: it is immeasurable in all senses and respects and hence, when ontically addressed (rather than addressed in terms of being a concept) it is nonquantifiable. As a thought experiment, try to imagine two ontically occurring metaphysical infinities side by side; since neither holds any delimitations (be these spatial, temporal, or any other) how would you either empirically or rationally discern one from the other so as to establish that there are two metaphysical infinities? In wordplay games, we can of course state, “two metaphysical infinities side by side” but the statement is nonsensical. More concretely, ontic nothingness, i.e. indefinite nonoccurrence - were it to occur (but see the paradox in this very affirmation: the occurrence of nonoccurrence, else the being (is-ness) of nonbeing) - is one possible to conceive example of metaphysical infinity. Can one have 1, 2, 3, etc., ontic nothingnesses in any conceivable relation to each other? (My answer will be “no” for the reasons just provided regarding metaphysical infinity. However, if you believe this possible, please explain on what empirical or rational grounds.) — javra

I don't wish to be mean, but this strikes me as complete word salad. The Wayans brothers (In Living Color) used to do a skit where two self-educated street preachers have a nonsensical conversation by stringing together unrelated words. Reminds me of that.

The bolded sentence ends in a question mark, so I assume its a question. And it seems to represent the crux of your argument. Can somebody translate?

... and that of “non-metaphysical" (aka, countable, mathematical) infinity (such as can be found in a geometric line of infinite length), — javra

Um, the points of a line may be put into one-to-one correspondence with the set of real numbers, which Cantor proved to be uncountably infinite in 1874. In fact, the points in a tiny line segment are uncountable.

I'm unsure why you're hung up on causal determinism. Do you think two points in the plane cause a line to be? I.e., the line was not there before? How else is countable infinity determinate? Because the act of counting gives us the set in its entirety? (OK, try it - count to infinity. We'll wait.)

Seeing how I’m having a hard time in even getting people to understand the problem, my only current conclusion regarding this problem is that it’s so dense that I needn’t concern myself with it when specifying metaphysical possibilities of determinacy. — javra

You seem genuinely interested in the topic. Depending on your math background, you could try to find a source that discusses the concept of infinity in math that you can use to begin to understand it. You could google texts on Set Theory for beginners, or find nice presentations on Youtube (this might be a good starting point).

If we admit that a line is not ontically determinate, then I suppose it's ontically indeterminate. I think your problem lies in equating "indeterminate" with "vague". One may draw part of the line in a specific location, just not the line in its entirety. Is this what you're looking for?

If a line (not a line segment) is ontically determinate, I assume you can draw it in its entirety. No?

I can't. Can you?

...with the supposition that any of this makes sense. — Banno

Oh, Banno. You're ruining our fun.

determinacy/indeterminacy and finitude/infinitude are defined by the ontic presence or absence of limits/boundaries — javra

I would like a better definition of determinacy. You seem to be implying that the line is determinate because the line exists in its entirety in the plane. Is this correct?

its width and shape is subject to fully set limits or boundaries, thereby endowing the geometric line with a definite uncurved length. — javra

So width is length?

And what is "uncurved" length?

Still eager to learn.

Yet the infinite length of a geometric line is definite, — javra

Can you elaborate? Do you mean that the line is measurable?

I know so little about math, but I'm always eager to learn.

I was thinking they could have named it after the old Saturday Night Live skit "Debbie Downer".

I don't trust that Bob. No one can be that nice. The weasel's up to something. When no one else is around, he probably leaves his toenail clippings on the rug, and cheats at solitaire.

Besides, someone's gotta stand up for the Larrys of the world. Poor schmoes. It's not his fault - he can't help it his parents were Republicans.

It's clear that the virtuous and good Larry is a much better person than lay-about, tree-hugging Bob. Anyone can see that!

You truly know nothing about math. Do yourself a favor and look up a term before spouting off nonsense about it.

Mapping is a commonly used math term. A reflection is a type of mapping.

From Britannica (online) - although you can find similar definitions in many places :

Mapping : any prescribed way of assigning to each object in one set a particular object in another (or the same) set. Mapping applies to any set: a collection of objects, such as all whole numbers, all the points on a line, or all those inside a circle. For example, “multiply by two” defines a mapping of the set of all whole numbers onto the set of even numbers. A rotation is a map of a plane or of all of space into itself. In mathematics, the words mapping, map, and transformation tend to be used interchangeably.

You revel in your willful ignorance of math. You puff out your chest and promote yourself as the folk-wisdom hero who must bring down all mathematical evil-doers. How's that going for you? Have you ever taken my advice and sent your math musings off to prestigious journals for publication?

A being cannot be both omniscient and omnipotent. Being omnipotent means having free will to choose one's action at all times. Being omniscient means knowing what every future choice will be. Can't have both.

Well, unless you allow for a universe that bifurcates (multi-furcates?) at each choice. So, omnipotence OR omniscience OR the multiverse (if both).

Or maybe gods don't exist (my vote).

If there was a spot on the reflective surface, a point on the plane, which reflected back on itself, "itself" being a point on the reflective surface, would reflect it again. back on itself, and again and again. — Metaphysician Undercover

Where do you get this from? This is not how mappings work. They don't repeatedly occur. Sure, two mirrors angled just so with space in between them will create an infinite regress, but that is not what is happening here. Here, we're considering a single plane of reflection, and a single reflection (a single mapping). You've invented a situation that doesn't exist.



I imagine you're a wonderful person, so it pains me to have to say this : usually, discussing math with you is like discussing the phases of the moon with a flat-earther. You really have no idea what mapping, or inverse, or almost any other math term means. And you have no interest in learning.

What's truly odd is that you're lack of understanding is at the most basic level. You stumble on understanding simple facts about the integers and zero. The Chinese and the Hindus understood the nature of zero thousands of years ago, and even late-to-the-game Europe has known about zero at least since Fibonacci's Liber Abaci. No one debates this stuff anymore.


Hey, I'm going to throw some group theory at you. Try not to let your head explode.

A group is a set of elements ($G$) and a binary operation ($*$) on those elements that satisfies 4 conditions :

• The set of elements is closed under the binary operation (i.e., if $a \in G$ and $b \in G$, then $a*b \in G$)
• The operation is associative on the set of elements (i.e., $a*(b*c)=(a*b)*c$)
• An identity element exists in the set (i.e., there is a unique element $e$ in $G$ such that $e*a=a*e=a$ for all $a \in G$)
• Every element of the set has an inverse (i.e., for all $a \in G$ there exists $b \in G$ such that $a*b=e$)

Theorem : In a group, the identity element is its own inverse.
Proof : By condition 3, $e*e=e$ (given any element - including the identity element - performing the binary operation with that element and the identity will result in the given element). Thus by condition 4, $e$ must be the inverse of itself.

In the discussion we've been having, the integers (positive, negative, and zero) are clearly a group under addition, with the identity element being 0. So by the theorem above, 0 is its own inverse.

The analogy really does not work Real Gone Cat. A reflection is light rebounding back off a reflective surface, which you represent as a plane. If there was a spot on the reflective surface, a point on the plane, which reflected back on itself, "itself" being a point on the reflective surface, would reflect it again. back on itself, and again and again. This would create an endless back and forth between the spot and itself. This is like having two mirrors in front of each other, accept that your proposal builds this right into the single plane, or reflective surface..

If such spots existed on the surface, each spot would effectively annihilate the capacity of the mirror to properly reflect at that point because the reflection would get absorbed into the infinite back and forth with itself. So if the rules of mathematics allow that zero "maps to itself" in this way, this would effectively annihilate the integrity of the concept "zero", as such a reflective surface (separation) between positive and negative, just like a spot on the mirror reflecting back and forth on itself would absorb the light and not reflect outward, ruining the integrity of the mirror as a reflective surface. — Metaphysician Undercover

This is entirely your own invention. Give one citation to support this. Just one.

This is an example of you digging in your heels. You're so math-phobic you have to invent concepts out of the blue to justify your stance. But "you know what you know".

If we allow that "zero" implies both positive and negative (in a self reflecting way) in common applications, instead of neither (as we actually do), this would destroy the integrity of "zero" — Metaphysician Undercover

Point out where I said zero is both positive and negative. Here, let me help you :

By the way, 0 is neither positive nor negative, so let's drop that nonsense now. — Real Gone Cat

Another jokester. Lord, I must have left my sense of humor in the car.

Hee hee, very droll. Lost my sense of humor for a minute there.

Okay. But isn't that just to say either there's no math that defines a value for it or that you're unfamiliar with math that does.

To just say, nope, is like saying negative numbers don't have square roots, or, for that matter, that 2 doesn't. — Srap Tasmaner

Are you arguing that some undiscovered branch of math looms out there waiting to give new meaning to well-established results? Possibly but very unlikely.

By the way, no one says negative numbers don't have square roots. What they DO say is that negative numbers don't have real square roots.

Actually, $0^0$ is an interesting case. Cauchy made the claim that it was indeterminate in 1821 and most calculus texts today echo that idea. Certainly the limit as $x\rightarrow0^-$ is undefined. But there are some contexts in which assuming $0^0=1$ helps to facilitate certain theorems.

Don't know about quantum physics having multiple types of 0, but doubt it.

Why not use a pair of these? $0 +$ $0 -$ $0^+$ $0^-$. They are commonly used in math. You could come up with the first being infinitesimals just to the right of zero, etc.
There are your "opposites" of zero. — jgill

No good. $0^+$ and $0^-$ are used in limit notation to indicate one-sided limits but have nothing to do with opposites.

You're the one who seems to be insisting that the rules you've mentioned have no use even within the realm of math itself. — frank

Citation please.

Aah, so math only has meaning if applied. Then why have you spent an entire page exchanging banal musings about the nature of 0 with Srap and others?

Hmmm, let's see ...

Farmers, doctors, lawyers, scientists, mathematicians, philosophers, ... Which of these has the least practical use? Which could society lose without blinking?

Then I take it you don't recognize pure math as having meaning. I wonder what this implies for philosophy?

Why do you need an application? That was not the question being addressed. Most of your posts address the nature of 0 and mine have as well. What are you aiming at?

I was addressing the idea that 0 cannot be across from itself. Now you want applications? I don't get you at all.