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
If "a" refers to "b" , doesn't this imply "b" refers also to "a". — Wittgenstein
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
If counting is something we learn, then counting is something we find."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" has the superficial grammar of a noun, but this is misleading. — Banno
Rather "1" is to be understood through its role in — Banno
Are letters objects? Are ink scribbles on paper objects? Are symbols objects?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? — Harry Hindu
"1" has the superficial grammar of a noun, but this is misleading. — Banno
So the extension of a set is the actual items in the set. — Banno
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?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 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
"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.
"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
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
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