Quantity without numbers
A perennial problem in the philosophy of mathematics concerns the nature of numbers: what are they exactly? Are they abstract entities, completely unlike all physical things? If so, how do we come to have knowledge of them? Many mathematicians claim that numbers are merely sets with certain properties. If so, then which sets, and why? No matter what numbers are claimed to be, there seem to be deep problems.
In thinking about these questions, I have come to a different conclusion. Numbers aren't things at all. They don't 'exist' in an abstract way or otherwise. When someone says 'I have many marbles', we don't claim that 'many' is a thing. Why then, if someone says 'I have two marbles' need we ask what 'two' is? Instead, when someone talks of having n things, for some particular value of n, we can always paraphrase this away. For example, 'I have a marble x and a marble y such that x is not the same as y and for any marble z that I have, z is either x or y'. The paraphrase is cumbersome, but shows that in such phrases the number term is dispensable.
[Philosophers of mathematics will note that Frege considers a similar approach, but gets into trouble with the 'Julius Caesar Problem' precisely because he wants numbers to be things. On the other hand, I consider it to be a virtue of my theory that numbers are not things.]
Similarly, we can paraphrase statements where the number appears as a noun. For example, if someone says 'Eight is even', we could instead say 'For any property, P, that applies to eight things, there are further properties F and G such that anything with P has either F or G but not both, and there is a one to one correspondence between those things that are F and those that are G'. Note that the only usage of 'eight' in this paraphrase was as an adjective, which can be paraphrased away using the first method.
So it appears that we can paraphrase away all particular natural numbers whether used as adjectives or as nouns, but what about universal statements, such as 'the sum of two odd numbers is even'. It turns out that by a further technique even these statements can be paraphrased so that no reference to numbers is required. Indeed, this can be done by taking the famous Peano Axioms of arithmetic and translating them so that they don't refer to any numbers. Instead of saying that zero is a number they say that there is a property that applies to nothing. Instead of saying that each number has a successor, they say that for each property there is a property that applies to an additional thing, and so on for the other axioms. Quantification over numbers is replaced with quantification over properties. Anything that you can do with 'seven' I can do with this technique using a property that applies to seven things. Does this mean that this property is seven after all? No, for the paraphrase is a general claim, and will work for any property that is satisfied by seven things.
It is worth noting that all of this could be done with sets too, but I find properties more basic. With this route, second order logic (plus a principle that there are infinitely many things) suffices to generate arithmetic.
I am currently writing this up as a paper for publication.