Hi Isaac! Thank you very much for your comments. The incompleteness of mathematics fits very well with a phenomenological picture (Gödel was very sympathetic to a realist phenomenological view). Formal mathematical systems on my view are analogous to particular angles from which an empirical object is seen. There’s always more to my coffee cup than what I see from one particular angle; there’s always more to a mathematical object than can be proved in any formal system. The analogy is of course more complicated than that, because establishing that you’re dealing with the same mathematical object (the same abstract structure) as defined in a different formal system is rather tricky. This is one of the places that category theory comes into play, showing us how different lower-level mathematical objects instantiate the same abstract structure (the same higher-level mathematical object).
As to any consensus in the philosophy of mathematics, you can’t get philosophers to agree about whether or not chairs exist or what “is” means. My view is fairly unusual in that most philosophies of mathematics either start with mathematical objects floating in the ether (Platonism) and then struggle to explain how we can know them or start with formal system described simply as the blind manipulation of symbols (formalism) and then struggle to explain how these manipulations have any relation to the world. Naturally my position is that you can’t start from these extremes and then try to explain the connection, you have to start with the connection (as it is given in experience) and then you can make sense of the objects and the formal systems that realize them.
As for next year, I am still applying to programs in the UK and Germany. Hopefully I will be able to attend a program in logic and the philosophy of mathematics in order to continue my interdisciplinary research, but I am also applying to general philosophy and mathematics programs. In short, nothing set in stone, but hoping for the best. I am particularly interested in learning more about the category of all categories (which I discuss in the conclusion of my IS project) as well as making sure I have a firm basis in axiomatic set theory.

Hi Isaac! Thank you very much for your comments. The incompleteness of mathematics fits very well with a phenomenological picture (Gödel was very sympathetic to a realist phenomenological view). Formal mathematical systems on my view are analogous to particular angles from which an empirical object is seen. There’s always more to my coffee cup than what I see from one particular angle; there’s always more to a mathematical object than can be proved in any formal system. The analogy is of course more complicated than that, because establishing that you’re dealing with the same mathematical object (the same abstract structure) as defined in a different formal system is rather tricky. This is one of the places that category theory comes into play, showing us how different lower-level mathematical objects instantiate the same abstract structure (the same higher-level mathematical object).
As to any consensus in the philosophy of mathematics, you can’t get philosophers to agree about whether or not chairs exist or what “is” means. My view is fairly unusual in that most philosophies of mathematics either start with mathematical objects floating in the ether (Platonism) and then struggle to explain how we can know them or start with formal system described simply as the blind manipulation of symbols (formalism) and then struggle to explain how these manipulations have any relation to the world. Naturally my position is that you can’t start from these extremes and then try to explain the connection, you have to start with the connection (as it is given in experience) and then you can make sense of the objects and the formal systems that realize them.
As for next year, I am still applying to programs in the UK and Germany. Hopefully I will be able to attend a program in logic and the philosophy of mathematics in order to continue my interdisciplinary research, but I am also applying to general philosophy and mathematics programs. In short, nothing set in stone, but hoping for the best. I am particularly interested in learning more about the category of all categories (which I discuss in the conclusion of my IS project) as well as making sure I have a firm basis in axiomatic set theory.

That should be (your) not (you’re). Type too fast and you will make silly mistakes!
I guess I also have a follow up question, as well. Do other mathematical philosophers agree with your view of mathematics through a phenomenological lens? Or do other researchers feel that perhaps there is a more appropriate way to view the connection of abstract objects to the world around us? Perhaps there is no consensus at all.

That should be (your) not (you’re). Type too fast and you will make silly mistakes!
I guess I also have a follow up question, as well. Do other mathematical philosophers agree with your view of mathematics through a phenomenological lens? Or do other researchers feel that perhaps there is a more appropriate way to view the connection of abstract objects to the world around us? Perhaps there is no consensus at all.

Thanks! That means a lot to me. I wish the best for you as well.

Thanks! That means a lot to me. I wish the best for you as well.