So if numbers are an aspect if counting, and one cant count to infinity, then finitism.
— frank

That's a bit too fast, but in being wrong, might be the gist of what is going on. It's worth talking to a child about infinity to see the change in thinking as they realise that for any number they construct, someone can make a bigger one; they say "a squillion billion", you say "a squillion billion plus one". Then the confusion when they begin to realise that "infinity plus one" is still infinity. The game changes before them. — Banno

The fact that someone can add one to some number in no way implies some notion of infinity. If anything, adding one to some number just produces a finite sum, not an infinite sum, hence my mention of Aristotle's actual vs potential infinity. Potential infinity is an idea that can never be realized as actual infinity.

"Infinity plus one" is incoherent. Infinity would already include all the ones, twos, threes, - everything. So by adding one to infinity implies that infinity wasn't infinite in the first place.

hence my mention of Aristotle's actual vs potential infinity. Potential infinity is an idea that can never be realized as actual infinity. — Harry Hindu

That's finitism I believe.

Now, if you could offer some examples of "language use" where words and numbers are not used to inform or communicate, and does not equate to just making scribbles and noises, then I would be interested in talking about those cases.
— Harry Hindu

When I'm at home alone, playing a game and losing, I often shout out "for fuck's sake". I'm not informing or communicating with anyone. It's an expression of frustration, much like laughing is an expression of happiness and crying is an expression of sadness. I wouldn't say that any of these expressions point to or are about anything (in the sense of reference). They may indicate something, but that's not quite the same thing – talking fast indicates that I'm in a hurry, but that doesn't mean that my words refer to the fact that I'm in a hurry. — Michael

Right, so in your example, you'd be just making noises with your mouth.

Like I said, any instance where you aren't using words to refer to something, or to inform you of something, then you're just making scribbles or noises, but then making scribbles and noises are themselves about, or can inform someone of something.

If I can determine that you are in a hurry by the way you speak, and not what you said, then if you had said, "I'm in a hurry" wouldn't that have been redundant being that you communicated you being in a hurry by the way you spoke? We often communicate without knowing it using body language. Talking is just another form of body language, of what certain bodily behaviors can be about. With words, we've simply added another layer of aboutness. Not only can I determine that you are speaking, and that you understand English, but your words are themselves about other things - another layer of meaning. I could tune out what you are saying and focus on your lingo and use of the language if that were my goal. Information is everywhere and is causal. The information I can ascertain from some effect (like your typed words) is the relationship that effect has with all the causes that lead up to it (like what you know and your understanding of the language you are using), it just depends on which set of causes we are focusing our attention on at the moment.

That's finitism I believe. — frank

Only that our thoughts are finite. We don't know if the universe is.

But then this brings up how our thoughts relate to the world. In thinking about infinity, do we really need to have infinite thoughts? In thinking about some thing, whether it be an apple or infinity, do our thoughts ever exhaust what it is that our thoughts are about? Is it preposterous to assert that in thinking about an apple, you exhaust everything about apples, and the same about infinity? If not, then is it necessary for thoughts to exhaust everything about some property or object to still be about those properties or objects? Even though our thoughts of apples might not exhaust everything that makes an apple an apple, apples still exist, right? So could it be the same case for infinity - that our thoughts about infinity don't necessarily need to infinite to be about infinity. It seems that is what thoughts are - a model of what it is that we are thinking about, and models don't exhaust what it is that is modeled, but still have a (causal) relationship with what is modeled.

Someone once told me that god is like a coffee cup, the mind picks it up and moves it around by the handle, never fully grasping it.

With an idea, we compare, contrast, measure, in short: relate it to other things. Every part of this process derives some portion of understanding.

The page you referenced quite explicitly sets out the difference between average velocity and instantaneous velocity. — Banno

I know there's a difference between "average velocity" and "instantaneous velocity" that's evidently obvious. However, "instaneous velocity" is still an average. It's just a different average from what is called the "average velocity". Here is how the page defines "instantaneous velocity":

"It is the average velocity between two points on the path in the limit that the time (and therefore the displacement) between the two points approaches zero."

Notice the word "average" there? I don't see why this is so difficult for you. Any determination of velocity, is necessarily some type of average due to the nature of time. It requires determining the difference between two distinct sets of circumstances to produce one result, called "the velocity". That is an averaging, coming up with one description from the two, you take an average between the two.

So the rule is that for every number, one can add one. The rule only generates one new number. One has to see the rule in a different way in order to understand infinity: imagine a number bigger than any number the rule could generate... — Banno

Do you see how this notion of infinity is inconsistent with constructivism? The bigger number referred to is not something which the human mind could ever apprehend, therefore it is beyond the capacity of understanding through constructivist principles. It's something which is simply stipulated, but never grasped therefore outside the range of intelligibility for constructivism, just like the "God" of the ontological argument, which is an inverted type of the same principle. "That than which nothing greater can be imagined", is a stipulation, which by the very nature of the stipulation cannot be grasped, because we can always imagine something greater. The same thing is the case with your "bigger" number, you are simply stipulating that no matter how big a number you can come up with, there's a bigger. The number you come up with is within the grasp of the mind, and comprehensible, the bigger number is always outside the grasp of the mind, therefore not comprehensible, and outside the principles of constructivism.

In short, you are suggesting that there is something (a number) which we can understand, which is outside of our range of understanding. The principle you propose is actually unintelligible.

Do you see how this notion of infinity is inconsistent with constructivism? — Metaphysician Undercover

Right, so the question that follows is: what happened so that we generally rejected constructivism? And what are the philosophical costs of having done so?

I thought @Banno had moved towards solving his own problem and was about to congratulate him. Have I got this wrong?

If "1" refers to an idea, then it is an idea shared. Else your idea of 1 would not be the same as mine.

So what sort of thing is that? — Banno

So,it's a shared idea? What's the problem? Don't we all have (more or less, that is sufficiently) the same experience and understanding of quantity?

My thought was that "1" might refer to "1". A circularity. Not that "1" might refer to 1. — ZzzoneiroCosm

I don't see how '1' could refer to '1' ; it is the referrer. not the referent, according to any logic that makes sense to me. I don't see a problem with '1' referring to 1; the apparent problem arises because we want to reify 1, and be able to say just what it is. I think it's just a conceptual illusion of substance.

A sort of thing that is. — ZzzoneiroCosm


Abstract object. — frank

I would rather leave the word "object" out here, because it invites reification. Better just say "abstraction" instead.

The prevailing view among mathematicians is that it's an abstract object. You can call it whatever you want.

The prevailing view among mathematicians is that it's an abstract object. You can call it whatever you want. — frank

If by "abstract object" you mean a platonic object, then I think you are drawing a long bow in claiming it is the "prevailing view" among mathematicians. Even if we granted that, though, this being a philosophy forum, the prevailing view among mathematicians would not be as relevant as the prevailing view among philosophers.

What do you think "abstract object" could mean outside the context of platonism?

You might find this article helpful.

Thanks. I think there's an SEP article on abstract objects if you're interested.

I don't see how '1' could refer to '1' ; it is the referrer. not the referent, according to any logic that makes sense to me. I don't see a problem with '1' referring to 1; the apparent problem arises because we want to reify 1, and be able to say just what it is. I think it's just a conceptual illusion of substance. — Janus

I don't see how "1" can refer to 1 in the same simple way (let's say) "justice" or "beauty" refer to justice or beauty. There's almost no information in the phrase: "1" refers to 1. In my mind, the second 1 is begging for quotation marks. Translated from numeralese, it would read: The numeral one, barring some preliminary qualification, refers only to the numeral one.

Clearer to say "1" refers (not to 1 but) to the concept of the singular or of the first.

At any rate, I don't find all of this very useful or interesting. It's obvious enough that "1" refers to something.

Thanks, I'll give it a read.

Clearer to say "1" refers (not to 1 but) to the concept of the singular or of the first.

At any rate, I don't find all of this very useful or interesting. It's obvious enough that "1" refers to something. — ZzzoneiroCosm

Yes, I agree that it is clearer; but I would tend to say that one is the concept of the singular or the first. In the usual philosophy of language 'x' ( any 'x') refers to x. If x is an idea then it can be expressed like this: 'x' refers to <x> where the <> stipulate that what they contain is an idea.

If I think about justice, for example, I can ask whether justice is an idea or an activity or action. So, 'justice' refers to justice, where justice is thought of as an activity, but I might write: 'justice' refers to <justice> if I want to say that the word justice refers to the concept of justice.

That makes sense.



If I compare:

"Tree" refers to tree.

and

"That tree" refers to that tree.

- the first phrase makes almost no sense to me and the second phrase is fine. (Maybe the first refers to the concept of a tree, but it isn't really clear.)


Repeated with "1":

"1" refers to 1.

and

"That 1" refers to that 1.

Neither of these make much sense to me.

"That 1" refers to that 1.

Neither of these make much sense to me. — ZzzoneiroCosm

How about "'that one' refers to that one"? Would saying "'one' refers to one" be different than saying "'1' refers to 1"?

How about "'that one' refers to that one"? Would saying "'one' refers to one" be different than saying "'1' refers to 1"? — Janus

I think I see how it works.

The fact that someone can add one to some number in no way implies some notion of infinity. — Harry Hindu

Sure. But that fact that someone can add one to any number does.

Notice the word "average" there? — Metaphysician Undercover

Yeah. See how delta-t becomes zero? So your average is a division by zero.

I don't know how to help you see the error you have made; my replying to you just backs you further into a corner.

The bigger number referred to is not something which the human mind could ever apprehend — Metaphysician Undercover

But that's not right; mathematicians, even those in primary school, do apprehend infinity in their considerations. There are whole books about it.

Further it is clear that infinity of one sort or another is easily constructed from a few simple considerations.

And i think that is an end to this discussion.

But that's not right; mathematicians, even those in primary school, do apprehend infinity in their considerations. There are whole books about it.

Further it is clear that infinity of one sort or another is easily constructed from a few simple considerations.

And i think that is an end to this discussion. — Banno

:up:

I've been reasonably satisfied with the replies of those who seem to know what they are talking about - @Sam26 and @jgill in particular.

I've come to realise that what I have been calling constructivist maths is not quite what is more generally called constructivist maths.

I reached a conclusion fairly early, and it seems that the talk of extension and intension in Wittgenstein’s Philosophy of Mathematics was misleading - whether in being eccentric or just wrong, I'm not sure.

It's something I will probably come abck to, but for now I am content.

OK. Was there a point on which we disagree? Perhaps not.

I don't know; I was merely indicating my agreement with the point you made in the passage I responded to regarding the genesis of the idea of infinity. I had already made pretty much the same point myself in this thread.

Well, damn. That don't make for a discussion if all we do is agree.

True, I guess, but I can't see much point in feigning disagreement. :grin:

It helps me to procrastinate...