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I understand numbers to be defined as objects defined to have certain convenient properties in relation to certain operations. It is very surprising that the exact same group objects should be applicable to the modelling of such seemingly disparate concepts as length, area, degree, cardinality, time, wavefunction, etc. I also find it surprising that the qualities of numbers, which appear quite contrived, should correspond to anything real in the first place. How can I make the relationship between these concepts and the concept of number more intuitive and expected?

tom894
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    I would have to disagree. I do not find length, area, volume, cardinality to at all be disparate nor do I think the properties of numbers are at all contrived. I can count how many sheep or apples I have and if its even I can evenly divide them between my two sons. Seems perfectly intuitive. Even fractions can come about this way, and once you start thinking about debt, negative numbers seem perfectly reasonable to me as well. And by measuring real things about circles, we discover $\pi$, by considering continuously compounded interest, $e$, etc. – Nap D. Lover Sep 30 '23 at 18:04
  • @NapD.Lover Appealing to situations where numbers are applied intuitively just tells us that we have an intuition for numbers and that their application is intuitive, it doesn't at all make intuitive why there is this application and why there is this intuition. – tom894 Oct 01 '23 at 00:24
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    Numbers are "generic" names for objects: we use them to "count", i.e. to differentiate individuals that are "identical" and thus we cannot differentiate them using proper names, like e.g. coins, sheep. – Mauro ALLEGRANZA Oct 03 '23 at 06:47

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I wonder where the puzzlement starts?

  1. Are you puzzled that we can use the quantifiers when talking about different kinds of things? Are you puzzled that e.g. we can use $\exists xWx$ to say there is a widget or to say that there is a wombat, depending on how we interpret '$W$'?
  2. If that's not puzzling, are you puzzled by the use of the identity predicate? Are you puzzled that e.g. we can use $\exists x(Wx \land \forall y(Wy \to y = x))$ to say there is exactly one widget or to say that there is exactly one wombat, depending on how we interpret '$W$'?
  3. If that's not puzzling, is it puzzling that we can similarly express e.g. there are exactly three widgets or exactly three wombats using the same topic-neutral apparatus? -- $$\exists x\exists y \exists z([Wx \land Wy \land Wz] \land [x \neq y \land y \neq z \land z \neq x]\\ \land \forall w(Ww \to [w = x \lor w = y \lor w = z]))$$
  4. But if there is nothing puzzling about the construction "There are exactly 3 $W$s" being topic-neutral, being applicable to different sorts of things, would there be something puzzling about the construction "The number of $W$s is 3"? being topic-neutral? Ok, we now we are moving from numerical quantifiers to numbers-as-objects (though of course we want there to be some sort of equivalence between "There are exactly 3 $W$s" and "The number of $W$s is 3"). Yet it is very difficult to see why we should be unworried that numerical quantifiers can be topic-neutral, yet think that is surprising that numbers as objects can be applicable in a topic-neutral way ....

The point that we can talk about the number of $W$s of lots of different kinds of thing is something that needs to be accommodated by any account of arithmetic. For one way of pursuing this thought, you might be interested in looking at this encyclopedia piece on Frege's extraordinary work: https://plato.stanford.edu/entries/frege/

Peter Smith
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