It makes doing mathematics like walking across a minefield! — jgill

It just means there are conundrums lurking whether you embrace constructivism or reject it.

What differentiates 1 from 0? — Outlander

What you do with one is not the same as what you do with zero.

Which gives it a value higher than nothing?

Needless to write 0 and 0 is 0 while 1 and 1 is 2. Has to have some value?

Which gives it a value higher than nothing? — Outlander

Yep. Your point?

Maybe I haven't been following the discussion but, therefore "1" has to refer to something? Not "anything" I suppose but it certainly doesnt refer to nothing as in 0.

so yours is a transcendental argument... something like:

One is greater than zero.
The only way one could be greater than zero is if the thing one refers to is greater than the thing zero rferes to...

Hence one must refer to something.

Is that your argument?

Sort of. Folks call zero a placeholder. It's the youngest of all numeric concepts, if I'm not mistaken.

If we are referencing something, using numbers. Zero does refer to something, rather lack of it. One is one unit, item, whatever the thing is. Two is two so on and so forth. Right?

Well transcendental arguments are notorious in that they rely on "the only way...". My point is that one is greater than zero not because of what it refers to, but because of the way we treat it. Would you rather one dollar or zero dollars? And hence that "the only way..." fails because there are other ways.

My argument, so far as I have one, is that the extension of "one" - the bunch of things to which "one" might refer - is each and every individual; and hence, the extension of "one" is anything. And if that is so, then one is not differentiable from anything else; it makes no sense to talk of the extension of "one". The corollary is that since numbers are all built form one, none of them refer.

But you seem at the end of your last post to offer another argument; that "zero" refers to nothing, and hence is another example of a number that has no referent.

My argument, so far as I have one, is that the extension of "one" - the bunch of things to which "one" might refer - is each and every individual; and hence, the extension of "one" is anything. — Banno

It's used as an adjective, so it's like "blue" or "fast." I have one elephant. It's modifying a noun. So what's the extension of an adjective?

It can also be a noun: "One is the loneliest number." The reference us a number.

The extension of "blue" is each and every blue thing - do you agree?

Hence it makes sense to talk about blue things because there are things that are not blue; there is stuff that is not part of the extension of "blue"

But if the extension of "one" is each and every individual thing, then there is nothing outside of that extension.

Yes that's a pun on "nothing".

The extension of "blue" is each and every blue thing - do you agree? — Banno

Blue can be used as a noun, as in "cobalt blue". We're contrasting a hue or shade with others.

When "blue" is used as an adjective, it's reference doesn't appear to be a spatiotemporal object. Maybe it's not the kind of thing that has a reference.

Would @fdrake know?

Would fdrake know? — frank

You are asking fdrake if he knows if you agree with the standard definition of extension in logic?

Sorry, what us the standard view of the extension of an adjective?

...presumably each of the things that satisfy that adjective.

You're presuming the standard view?

Yep. You?

Do you suppose there's an SEP article that clarifies the standard view?

Why not take a look?

Sure. Manyana.

Mathematics, properly understood, is the theory of everything.

The proper understanding of math derives all numbers, mathematical operators, and all the other mathematics, from the starting symbol 0.

Then as by rules most parsimonious, there would be an ordering of mathematical structures, in regards to the zero.

The start would be 0. Then to derive the 1 from the 0, the 0 is rewritten as a 1. The rewrite principle is like when the same information can be rewritten to harddisk, and to a dvd, and to RAM, and it's still the same information, but rewritten in a different form.

So the 1 is essentially a rewrite of the 0. Now we also get the boolean operator, because 1 being a rewrite of 0, means they have boolean interchangeability.

Now we have the mathematical structures of 0, 01, 10, 00, 11

0, 01, 00, and 10 would all total 0, because of the boolean operator making the 01 and 10 also total zero. But there is no boolean operator for 11, making it not total zero, which is totally uncool.

Therefore logic must respond to make it total 0 zero again. So logic responds to the 11 with a 00. And all is parsimoniously total zero again, and we get additional mathematical structures. And so on.

And it should be shown to be the case that in physics, the physical 0 and 1 in isolation, would have boolean properties, that this would be some kind of phenomenon of physics.

And the main laws of physics, and constants, would be apparent in the ordering of the mathematical structures in respect to the zero.

And the human mind, and the DNA system, and the universe, would all be shown to have the same fundamental mathematical ordering. But no the sequence of CATG, or where the planets and stars are, and what pictures someone dreams of. That is not fundamental.

What is this a picture of? — Banno

I'd call it a video loop. What does it have to do with a number line, or counting? A loop is not a line, and if it's counting anything it's counting the same things over and over again.

Mathematics, properly understood, is the theory of everything. — Syamsu

:cool:

The extension of a property is the collection of objects which satisfies the property. "is an object on my table" has extension "my laptop, a half litre coffee mug, a heat mat, a candle holder, a plastic water jug, a 2 factor id device, an unplugged microphone, a computer mouse and 2 boxes of oral nicotine pouches/snus".

https://www.britannica.com/topic/intension

In terms of your discussion with @Banno, extension and intension are different ideas in general than referent, except maybe in the case where you're already dealing with a word or phrase which is being used to refer, like the proper noun "frank", with my intension you.

Oh, I see. Thank you!

So essentially any number would not refer to anything either? If so what does zero refer to? What differentiates 1 from 0? — Outlander

1 can count the 0. 'One zero'.

Huh. Suppose it can. :D

You can create the number line with the null set. Let {0} = the null set:

{0}
{0}{0}
{0}{0}{0}
{0}{0}{0}{0}...etc

= 1, 2, 3, 4...:

You can create the number line with the null set. Let {0} = the null set:

{0}
{0}{0}
{0}{0}{0}
{0}{0}{0}{0}...etc

= 1, 2, 3, 4...: — EnPassant

And then come the fractions . . .

That's another kettle of fish!

1 can count the 0. 'One zero'. — EnPassant

'One infinity'.