What Numbers Cannot Be, and the 'Leininger Postulate'

Benacerraf has been one of the more adamant critics of Platonism in Mathematics. His most famous arguments focus on the epistemological nature of such abstractions, and the seeming over-determination of a set-theoretic description of the natural numbers. Dustin posed an argument that seems to sidestep the problems...or does it? Most of these arguments are relatively easy to explain, so I hope I don't mess this up...

Epistemological Argument:
1) Numbers are Platonic entities (and as such, atemporal and non-spatial) (Assumption for reductio)
2) The causal theory of knowledge is correct (Assumption)
3) We cannot causally interact with numbers, hence we have no knowledge of them (1,2)

If we do, as we commonly think, have knowledge of numbers, then it seems that something assumed was incorrect. Critics of the argument took umbrage with (2), but the Argument can be accurately reformed for other epistemological theories of knowledge like contextualism and reliabilism. It seems then, that (1) might be the issue....

Over-Determination Argument:
We can define numbers in terms of sets in many ways (infinitely many really). For instance, one description identifies the null set (null) with zero, and then continues additively [null]=1, [[null]]=2, [[[null]]]=3, etc. Alternatively, we could also define numbers in the following way, null=0, [null]=1, [null, [null]]=2, [null, [null], [null, [null]]]=3, and so forth and so on. Obviously, they are isomorphic, and so calculations go through unabated, but meta-mathematical statements lead to inconsistencies. As Benacerraf pointed out, one description yields 2 as a member of 2, and the other that 2 is a member of 3. We can reasonably ask, which one is correct? Table this for now; and on to the next one...

Now Dustin is an impressive up-and-coming Physics student with a sharp intuitive instinct (Just to be clear, I am not being sarcastic. I often am of course, but not this time.).

Leininger Postulate:
It is generally accepted that mathematics can be derived from the natural numbers. All things considered, establishing the natural numbers would go a long way in supporting common sense mathematics. Moreover, additive properties can be invoked to develop all of the natural numbers from merely zero and one. Thus, establishing zero and one would, in effect, establish the natural numbers and subsequently, a great deal of mathematics. Zero and One, however, are intuitively what we mean when we say 'nothing' and 'something.'

So, what we seem to have here is two very different mental constructions from which all of mathematics can be derived. I think this is quite plausible, but I would love to hear some feedback from the club!

Pardon me for ignoring the ontology of numbers...

Here's what a contextualist might say:
- Take the actual world to be a centered possibility, and all counterfactual possibilites to be surrounding worlds.
- Our knowledge is determined by setting a boundary at which possibilites we're ignoring. So when you ask me how do I know that I'm seeing a zebra instead of a donkey painted in stripes, I'm ignoring that possibility because it's so unlikely; our world would have to quite different.
- The possibility in which logical and/or mathematical truths are otherwise are among the farthest from actuality, so we're pretty much always ignoring them.
- If we granted the possibility that mathematical truths aren't necessary, then we'd have to grant that just about anything is possible. And so I really don't know that the zebra isn't a donkey, or anything else for that matter. But we do know SOME things, so that can't be right.
- Alternatively, replace the centered possible world metaphor for the notion of a maximal epistemically consistent set of propositions.

Connection to Philosophy of mind:
If mathematical truths are necessary, then they're a kind of narrow content that is shared by anyone who thinks about them. So Twin Earth arguments don't work on mathematical beliefs. David Chalmers construes narrow content as "primary or epistemic intensions", which are functions from centered possible worlds (mentioned above) to referents. He then defines the A Priori as epistemic necessity: when the primary intension (narrow content) is true in all possibilities.

I suspect, though, that you'd be willing to grant that logic and mathematics are necessary in some sense... but don't refer to real things. So again, sorry for ignoring the ontology, just wanted to bring in these other perspectives.

Contextualism and Two Dimensional Semantics! *These are two of my favorite things*

The Contextualist narrative is always useful, but seems ineffective against Benacerraf's Epistemological Argument for the following reason: 1) Replacing 'Platonic entities' seems to require the comparable 'modal actualities' in which case the argument goes through. Of course, this was not intended, I am sure, to offset the argument, but I wanted to be sure that the reformulation is clear.

Now,

If we granted the possibility that mathematical truths aren't necessary, then we'd have to grant that just about anything is possible. And so I really don't know that the zebra isn't a donkey, or anything else for that matter. But we do know SOME things, so that can't be right.

I will take your bait. Assume mathematical truths are not necessary. Then they are not true in all possible worlds. Thus, in some world W 2+2 does not equal 4. I can conceive of such a world, though I am not entirely sure what this would entail. Does my lack of intimate knowledge of such a possible world mean that it is 'the farthest from reality.' Possibly. This cannot indicate that such a world is not possible, however, as I often have a lack of epistemological certainty about beliefs and what such beliefs might entail (this was one of the motivations for contextualism) In fact, as you point out, this seems to mean that just about anything is possible, but I am unclear as to how this would entail that the zebra isn't a donkey...I have attempted here to follow the consequences of non-necessary mathematical truths in an effort to find a contradiction (as indicated in the last line of the above quote), but I am afraid that I do not see where a contradiction emerges.

Of course, choosing not to ignore worlds which lack mathematical truths might not work well with disguised zebra, but then again, it might. If we are going to argue something like, "the law of the excluded middle does not hold" in this possible world which I am no longer ignoring, then it seems like we could have zebras and painted mules at the same time. But as I recall DeRose's zebra argument, the shift in context allowed me to shift my evaluation. So, I see a zebra, then you tell me it is a painted mule, and then accuse me of not knowing it was previously a zebra. I, correctly on this account, reply that I did know it was a zebra, but then with the added information (context) can see that I currently do not know that it is a zebra. With the above exclusion I might be able to respond, well, I did know it was a zebra, but now I know that it is both a zebra and a painted mule. This, while superficially problematic, seems reasonable...but I digress...

In fact, as you point out, this seems to mean that just about anything is possible, but I am unclear as to how this would entail that the zebra isn't a donkey..

You're right, which is why I think what I said doesn't really have any ontological force. The conclusion is supposed to be that allowing mathematical truths to be contingent would entail that we don't know anything, but we do know something, so they can't be contingent. I don't think it's a great argument because it's basically just saying that all knowledge--truth, really--presupposes the necessity of logic. So maybe there's whiff of circularity but I just wanted to fill in what an (amateur) contextualist might say. The point about having a zebra and a painted mule at the same time is nice too, didn't even think about that.

Epistemological Argument:
1) Numbers are Platonic entities (and as such, atemporal and non-spatial) (Assumption for reductio)
2) The causal theory of knowledge is correct (Assumption)
3) We cannot causally interact with numbers, hence we have no knowledge of them (1,2)

I think (2) and (3) are both problematic. (2) because it requires the causal/direct theory of reference. But numerals could be Fregean proper names, at least in Auerbach's view. Don't hold me to that though, just what I seemed to gather from barely skimming over his "Saying it with Numerals" paper. And (3) because modes of presentation and/or abstract objects aren't "physical", and if causation is a physical notion, then of course we won't causally interact with them in way remotely similar to how we interact with physical stuff.

Interestingly, I think this parallels your argument here:
Forcing the Burden of Proof

1) The causal theory of reference is an accurate portrayal of meaning (Assumption)
2) Consciousness is 'what it's like to be' from an arbitrary perspective (Definition)
3) Consciousness is reflexive and insulated (Problem of other Minds, 2)
4) Subjective personal consciousness has no causal link to external consciousness(es) (1,3)

So if mental representation/intentionality is a kind of abstract semantic content, then that would explain why we don't find beliefs or consciousness in brains. There's an intensional or "narrow" component to psychosemantics and numerals that we won't find anywhere in particular in the physical world. That doesn't mean other minds and numbers don't exist though.

By the way, I would love to check this out but I can't find it anywhere. Let me know if you do!