And yet they are countable. Look it up. — Banno

Denumerable, yes. Let's not mistake that for countable in the common sense of the term. I think that's where some of the confusion in this thread is coming from.

The natural numbers are also a proper subset of the rationals, but they're the same size. — Srap Tasmaner

You mean they have the same cardinality. Neither one really has a size.

The natural numbers are countable. — Banno

You couldn't finish counting them.

Still, forcing the unwieldy mass of rational numbers to line up single file to be counted was a master stroke. — Srap Tasmaner

It's just that the extension of the idea of the real numbers seems to be somehow bigger than the extension of the idea of the natural numbers. We could express that by saying it appears the set of natural numbers is a subset of the set of reals.

Neither set is countable, but that sense that one is bigger than the other was expressed in terms of cardinality.

Chickadees are dinosaurs, but they are a far tweet from T-Rex — BC

They're doing the best they can. There was an asteroid issue.

I like to emphasize that we are part of a continuum of life which has been created over a long period of time. Our evolutionary history is why "we are what we are" and every other species is what it is as well. — BC

About 99% of species that have existed on earth are now extinct. Do you have thoughts about the end of our species? I always thought it was kind of un-face-able, but do you feel like the possibility can be faced, and accepted? If we really are part of a continuum, maybe it's ok, because something else will take our place.

Being an object is a role in a language game, not an ontological status. — Banno

So the same thing will work for "abstract" and "platonism.". They're parts of a language game. You can't reject them without special pleading.

Godel said we perceive abstract objects. He would know.

L. Ron Hubbard didn't win a Hugo. In my collection, Asimov talks about the convention where his followers showed up demanding that he win. There was this weird tone in Asimov's writing when he talked about it.

It's a Hugo winner. It's in a collection of winners I have, edited by Asimov.

Why not? I have nothing in my pocket, therefore I have nothing. — Banno

It doesn't sound like you know what a set is.

You said this:

We've shown how quantification can be handled without invoking abstract objects at all — it’s rule-based, normatively grounded, and socially coordinated. — Banno

Now apply that strategy to the empty set. You'll find that you can't. Set theory is fundamentally platonic. Eject platonism, and you've ejected set theory.

A no to what? Set your account out. Say something. Do the work. — Banno

I've just been observing the different stances people are taking. The only book on phil of math I've read is Mary Tiles' book. After reading it, I realized the ways that set theory is conceptually objectionable, which might not be surprising since Cantor was a mystic, and his mathematical views were directly related to mysticism. Tiles doesn't campaign against set theory by any means, but she does leave the reader with the thought that we may one day rethink the whole thing. It may be that Aristotle was right after all.

Is that a no?

This is platonism. The abstract object is independent from minds, but accessed by them. — Metaphysician Undercover

Yes. I know. You pretty much came up with Frege's argument all by yourself. That's pretty cool.

We've shown how quantification can be handled without invoking abstract objects at all — it’s rule-based, normatively grounded, and socially coordinated. — Banno

Ok. I don't object to that. I doubt you can do the same thing for ZFC, though. So are you now suspicious that ZFC might be bullshit?

What of quantification? — Banno

What about it?

Perhaps you misunderstood...? — Banno

Could be.

You could say that. The point though is that if a numeral refers to a number which is an object, and that object is said to be an idea in someone's mind, then it would be a different object in each mind. — Metaphysician Undercover

An object in your mind is called a mental object. An object in your hand is a physical object. An abstract object is something that isn't physical, but it's not simply mental either.

The only way to assume that the numeral refers to the same object for distinct individuals, is to assume that the object is independent. — Metaphysician Undercover

That is correct.

So you've just completely changed your mind here. You were quoting ZFC as if it were scripture a few pages back. Then you were a deflationary nominalist. Now.. I have no idea what you are. I think you might be constructing your view as you go along.

Dude. According to that survey, a small minority think the foundation of math is set theory. There aren't a lot of experts in phil of math, but they would all roll their eyes at that. :grin:

That's permitted, under the rules... — Banno

You have the minority view, so you must be following the minority rules. :razz:

Elements in a language game can be things - because we quantify over them... all these numbers are even, all those numbers are prime. — Banno

You're trying to have your cake and eat it too.

Despite my obvious addiction, I am still functional. But it's going to be 36℃ today, 42℃ tomorrow, so productivity occurs inside or in the early morning. — Banno

-15C here. :confused:

What I said, is that if a numeral is taken to refer to an object, a thing called a number, that object must be a platonic object — Metaphysician Undercover

So math is just language games, right?

Wow! Being productive. :up:

There was a lot of strenuous protesting in this thread to the effect that infinity is a thing. Turns out you actually agree with Meta. Numbers aren't things. They're just elements of language games.

Why should I care? — Banno

Because you're procrastinating from whatever it is you're supposed to be doing right now?

So that's deflationary nominalism. It's a minority view.

Notice that this odd position is blandly asserted, not supported by any argument. — Banno

Frege had a pretty persuasive argument for it.

No, I've repeated this numerous times now, "space" is purely conceptual — Metaphysician Undercover

Are you a cartoon character? Do you know SpongeBob?

I might have said property - this counts as being mine. Basic idea is right. — Banno

That existed before money. They bartered. The problem was that corruption in bartering was rampant. They would put the good dates on the top of the caravan, and it was just mud-balls below that. It was so bad that it inhibited trade.

Money set trade free from corruption because it was these little pieces of gold which were stamped to assure a specific weight and purity.

Next came banking, which was mainly invented by the Italians. Now we have virtual money, which allows economies to grow past their present means. The human world as we know it today is a result of money and banking.

Math as we know it piggy-backed the development of money. Money, first invented in Lydia, was the first abstract object, typifying value, but not specifying the value of what.

So abstraction isn't a philosophical folly. It's the result of an astounding innovation.

I’m blocking a slide in what “exist” means here — Sam26

What are you blocking it with? Sentiment?

No, we don't understand either one. There are two distinct notions 'orthogonal' to each other plus a scale measure to create a rough translation from one language-less feeling to a public technical language that can be charted for other doctors to see. — magritte

You don't understand what a person is telling you if they say they're cold? Odd.

So it's not private. We understand what they're saying.

Can anyone other say how cold the feverish person is? — magritte

They say they feel cold. I understand what they're saying. What else is needed?

Yep. there is a difference between being cold and feeling cold, as is shown by the fact that we have that very grammatical structure. — Banno

Right, but when the feverish person says "I'm cold.". I don't tell her she's wrong. I believe she's telling the truth. The content of experience isn't private in the Wittgenstein sense.

That John and Jane disagree as to the temperature of the bath is not a fiction; it's something to be explained. This is lost in your account. — Banno

Definitely. It's not unusual that a person with a temperature of 103 F will ask for a blanket because they feel cold.

It might be said that they feel cold even though they aren't. But this is not to say they're experiencing an illusion. They really do feel cold. This is an excellent moment to see how words take on meaning in context.

Either way, my position is that "indirect perceptual realism is broadly equivalent to the scientific view of perception that subjects do not experience the external world as it really is" — Michael

But doesn't indirect realism suggest that there is no unmediated account? And therefore truth is going to be relative to life form?

For instance, cats see in conditions humans would call "darkness.". So is the room dark? It depends on what life form you are.

If there is no unmediated account, there's no unmediated truth.

@Punshhh

The local news station is broadcasting information about the 4th Amendment, which denies police the power to enter your home without a judicial warrant. They explain that ICE spokesmen have stated that they can enter homes without warrant, so the advice is to remain calm, don't open the door unless they show you a warrant, and basically be prepared to contact constitutional lawyers who are already preparing to bring cases to the Supreme Court. Strange times we're living in.