I read the article on Witts philosophy of mathematics some time ago, Wittgenstein rejects the set of real numbers or any sort of infinite mathematical extensions.
Extension (concatenation of symbols) will always be finite. Mathematics is all the combined knowledge of intentions and extensions and nothing beyond that. We can only understand infinity as an intention. There is an infinite possibility of natural numbers, not a set of infinite natural numbers according to Witt.

I can tell where he is wrong though, and it's here

Thus, Wittgenstein adopts the radical position that all expressions that quantify over an infinite domain, whether ‘conjectures’ (e.g., Goldbach’s Conjecture, the Twin Prime Conjecture) or “proved general theorems” (e.g., “Euclid’s Prime Number Theorem”, the Fundamental Theorem of Algebra), are meaningless (i.e., ‘senseless’; ‘sinnlos’) expressions as opposed to “genuine mathematical propositions.

"1" refers to "1" — ZzzoneiroCosm

No, it doesn't. It doesn't refer to anything. — Banno

I think you've made a use-mention mistake there.

1. 1 is greater than 0
2. There is a '1' in the previous sentence

The '1' in sentence 2 refers to the '1' in sentence 1, even if the '1' in sentence 1 doesn't refer to anything.

Another mistake you may have made is conflating numerals and numbers. Numerals refer to numbers, even if numbers don't refer to anything.

So when you say "'1' does not refer to anything" are you referring to the numeral or the number? If the former then I think you're wrong (the numeral refers to the number). If the latter then I think you're arguing against a position few would take (numbers, like tables and cars, are commonly thought of as referents, not referrers).

Folk seem to be missing the point: why does Wittgenstein's constructivism lead to finitism?

If "a" refers to "b" , doesn't this imply "b" refers also to "a".
Even if you do not explicitly state what the number "1" in the sentence 1 refers to. After you have established a connection to sentence 2, there is an implicit relation.

If "a" refers to "b" , doesn't this imply "b" refers also to "a". — Wittgenstein

1. I am a man.
2. The last word in the first sentence has three letters.

"The last word" in sentence 2 refers to "man" in sentence 1 but "man" in sentence 1 doesn't refer to "The last word" in sentence 2.

I needed a mathematician who might disagree with a constructivist approach to mathematics; — Banno

What, to explain,

why does Wittgenstein's constructivism lead to finitism? — Banno

? :chin:

One cannot physically list the integers. But in understanding the intension of "integer" we understand how to construct the extension... and in so doing it seems to me that we understand the extension to be infinite. — Banno

If in fact the concrete world is finite, acceptance of any theory that presupposes infinity would require us to assume that in addition to the concrete objects, finite in number, there are also abstract entities. [...]

Apart from those predicates of concrete objects which are permitted by the terms of the given problem to appear in the definiens, nothing may be used but individual variables, quantification with respect to such variables, and truth-functions. Devices like recursive definition and the notion of ancestral must be excluded until they themselves have been satisfactorily explained.
— Goodman and Quine, Steps Toward a Constructive Nominalism

"1" has the superficial grammar of a noun, but this is misleading.

Rather "1" is to be understood through its role in the process of counting. It is understood in learning how to count, not in pointing to individuals.

And of course this goes for other mathematical entities, too. They are things we do, not things we find. — Banno

If counting is something we learn, then counting is something we find.

If we aren't pointing at individuals when counting, then what are we counting, numbers or individuals?

"1" refers to the individual counted first among the counted.

Because you can't count to infinity.

"1" has the superficial grammar of a noun, but this is misleading. — Banno

"One" is a noun (or, more precisely, a pronoun) in the sentence, "I'll have one," where "one" is whatever you were referring to.

In the sentence, "I have one dog," "one" is an adjective and there is no referent to the adjective other than it being a descriptor of the dog. You reach a similar result with other adjectives, as in, "I have a happy dog." Happy is not a thing.

So, give me some examples of where "one" is superficially a noun where you're just not identifying "one" being used in the adjective case.

Rather "1" is to be understood through its role in — Banno

... in whatever the discourse. Charity and world-domineering ambition alike require translation between discourses, and agreement re ontological commitment (re, e.g., what "1" refers to). Pedagogy and practicalities require, instead, toleration of alternative systems.

1 points to X, as being thus 1X. 1 does not exist like Banno suggested.

What is the meaning of 1 if not pointing to X?

1 counted, is 1 count where X = count. 1 without X is 0.

, here is a quote from Wittgenstein in the Philosophical Remarks that might have some bearing on the subject.

"We can ask whether numbers are essentially concerned with concepts. I believe this amounts to asking whether it makes sense to ascribe a number to objects that haven't been brought under a concept. For instance, does it mean anything to say 'a and b and c are three objects'? I think obviously not. Admittedly we have a feeling: Why talk about concepts; the number, of course, depends only on the extension of the concept, and once that has been determined, the concept may drop out of the picture. The concept is only a method for determining an extension, but the extension is autonomous and, in its essence, independent of the concept; for it's quite immaterial which concept we have used to determine the extension. That is the argument for the extensional viewpoint (p. 123)."

We have a concept (a mathematical concept), and we use the concept to refer to things, but the things do not reflect the concept, i.e., it is not as though the concepts are intrinsic to the things. We group things together under the rubric of the concept, and we extend this concept to group other things under the same umbrella. "The extension is autonomous." The extension reflects a certain state-of-affairs that is brought under the mathematical concept.

The same would be true of a unit of currency. A dollar doesn't represent the value of any particular thing. So yes, "dollar" doesn't refer to the value of anything.

For instance, does it mean anything to say 'a and b and c are three objects'? — Sam26

Are letters objects? Are ink scribbles on paper objects? Are symbols objects?

Letters and numbers are each individual objects that can be counted. How many numbers are on this screen? How many letters? What are you counting when answering this question - objects or what?

Are letters objects? Are ink scribbles on paper objects? — Harry Hindu

You tell me, do you or we refer to marks on a piece of paper as objects? I think not. Some might say that they refer to objects.

No, it doesn't. It doesn't refer to anything. — Banno

Is that settled science?

"1" has the superficial grammar of a noun, but this is misleading. — Banno

This depends on how you understand "1". You can understand it as playing a role in counting, as you describe, in which case we can assign some sort of priority to it. But many modern mathematical axioms remove this priority, denying that priority, assuming that mathematics is something other than a tool for counting. Measurements are not all instances of counting because we employ negatives etc..

But you can also understand "1" as a fundamental unity, and this gives it a logical priority, as the designation of an object, a unity. This is required for all logical processes which proceed from the assumption of objects.

The two forms of priority are not completely incompatible though.

So the extension of a set is the actual items in the set. — Banno

The concept of "set" requires the assumption of objects. So set theory utilizes "1" as a unity, an object. This is an issue with set theory which I discussed with someone else in another thread recently. To assume that a set has extension is to assume Platonism, because it necessitates that a number is an object, being derived from that assumption. As I argued in that thread, "infinite extension", which is what conventional set theory allows, is incoherent, based in contradiction. An object, as a unity, being unbounded, is fundamentally contradictory.

What I found in that other thread, a conclusion you may or may not be interested in, is that there is an issue with the usage of the law of identity, in formal logic. The law of identity, as designed, is intended to assign uniqueness to an object. The law of identity as employed in formal logic designates a form of equality. Equality and identity are distinct ideas. Identical might be a specific way of being equal, but being equal does not necessitate being identical. In formal logic, the latter is assumed to be the case, that being equal is the same as being identical. So the law of identity, as originally formulated, is violated by formal logic, which employs a distinct interpretation (misinterpretation) of it.

No, it doesn't. It doesn't refer to anything. — Banno

What would your take on a formal semantics approach to 1's referent be? Like, taking it to be by definition the successor of 0, or the equivalence class under bijections of { { } }.

You tell me, do you or we refer to marks on a piece of paper as objects? I think not. Some might say that they refer to objects. — Sam26

What are marks on a piece of paper, if not marks of ink, or lead? Does ink cease to be an object when it gets transferred from the pen to the paper?

Symbols are objects used to refer to other objects. A stop sign is a sheet of octagonal shaped metal with red and white paint, that refers to the act of stopping one's vehicle.

To count correctly, you have to remember which individuals have already been counted. Numbers are placeholders for those individuals that have been counted, and represents the sequence in which they were counted, 1st, 2nd, 3rd, 4th, etc.

What would your take on a formal semantics approach to 1's referent be? Like, taking it to be by definition the successor of 0, or the equivalence class under bijections of { { } }. — fdrake

Different signifiers, same signified?

"1" has the superficial grammar of a noun, but this is misleading.

Rather "1" is to be understood through its role in the process of counting. It is understood in learning how to count, not in pointing to individuals.

And of course this goes for other mathematical entities, too. They are things we do, not things we find.

If 1 is to be understood through it’s role in the process of counting, wouldn’t it refer to a specific point in that particular sequence?

"1" has the superficial grammar of a noun, but this is misleading.

Rather "1" is to be understood through its role in the process of counting. It is understood in learning how to count, not in pointing to individuals.

And of course this goes for other mathematical entities, too. They are things we do, not things we find. — Banno

1 is simply an abstraction most beautifully captured with set theory as the property shared by sets of type: {a}, {#}, {£}, etc. (sets with only one element).

As to "1", I suppose we may proceed from self-identity?
Wherever we deem some such, like we talk about and point at things every day, we say there's (a quantity) of 1 of that, regardless of whatever exactly it may be.
Could include hypotheticals and whatnot, too.
It'll take some conceptualization to get 2 (of those), but we may instead have 1 of this and 1 of that, thus having 2 of this or that together.
So, in the abstract, 1 (just 1) would denote 1 of anything, without referring to anything in particular, but still exemplifiable.
Seems to be how we typically use 1 anyway, no?

We have a concept (a mathematical concept), and we use the concept to refer to things, but the things do not reflect the concept, i.e., it is not as though the concepts are intrinsic to the things. We group things together under the rubric of the concept, and we extend this concept to group other things under the same umbrella. "The extension is autonomous." The extension reflects a certain state-of-affairs that is brought under the mathematical concept. — Sam26

Thank you. I'm picturing this as direction of fit, as in Anscombe. That the number of things in the box is 3 is something we do; the direction of fit is from us to the world.

Add to that, that concepts are not things so much as a way of behaving; that is, concepts are best not considered as things in people's minds, but as ways of talking and acting. (compare street's recent thread on emotions as concepts)

Then we have a way of talking that goes "one, two, three" while pointing to each thing in turn. And we can use this way of talking to talk about lot of different things. And then we can talk about this way of talking when we find ourselves adding, then multiplying, then differentiating...

SO even though numbers are not things, we develop mathematics by treating them as if they are. And in the end they become thigns just by our having treated them as such.

So, the extension of "Sam26" is Sam26. The extension of "red" is each and every red thing. But the extension of "1"? Well it's literally every individual. And as such it seems to me, at least in my present mood, that the extension drops out of the game, and what we have is the intension, the rule, concept or game we play in counting.

And the consequence of that is that talk of extension in mathematics becomes fraught with ambiguity. Hence, Wittgenstein's argument that mathematical extensions must be finite, and hence his adoption of finitism, seems misguided.

:lol: This is both funny and profound.

Thanks. Yep.

Yep.

Their rejection of abstract entities remains unexplained - an intuition.

I hope I explained - in outline at least - that abstract entities come form our talking about what we do; so in talking about counting, we pretend that integers are real things, and this leads us on to more complex ways of talking about integers, and so a sort of recursion allows us to build mathematics up from... nothing.