I've been asking myself this and other questions in the field of philosophy of mathematics. Could we, if we were isolated from any kind of sensory experience, be able to learn mathematics?
This question has been moved from the math SE to the Philosophy SE.
Is math independent of our sensory experience?
Kants answer, in a sense, is that it is both dependent on sensory experience and also not. He claims that our intuition for space, through which we construct geometry, is a priori and thus independent of experience, but also synthetic, so that it is more than the rules of logic; he says that this is possible because our intuition for space is a neccessary condition to have any experience at all.
Frege, agreed with this for mathematics considered solely as geometry, but disputed arithmetic fits into what he calls Kants psychologism, perhaps a term that he picked up from Hume. Frege is a key figure in the early 20 Century project to reduce arithmetic to logic; and it is this thought that bypasses Kant, or so one supposes, because this would mean that arithmetic being solely based on logic cannot be synthetic, but must be analytic.
Could we, if we were isolated from any kind of sensory experience, be able to learn mathematics?
Thus, Kants answer is no for both arithmetic and geometry; and Frege says yes for arithmetic, and no for geometry.
As for learning mathematics - The SEP says on the Kantian philosophy of mathematics:
In a series of papers, Charles Parsons has argued that the syntheticity of mathematical judgments depends on mathematical intuitions being fundamentally immediate, and he explains the immediacy of such representations in a perceptual way, as a direct, phenomenological presence to the mind.
That is the abstract '2' as distinct from, say a concrete '2 bottles' or '2 books' that we might look at and perceive, is not abstract to our sensibility, it has a 'phenomenological' presence.
The hard work of learning mathematics is to synthesise these concepts so that the abstract concept has this actual sensual presence in the mind. One might say the moment of 'clarity' or 'illumination' is a spark given of by this act of mental synthesis. This is the beginning, the process and becoming of the mathematical Subject - subject as in subjectivity, not as in topic.
I believe the answer to your question would be no.
Sensory inputs are the only mechanism for input into the brain aside from evolutionary 'pre-programmed' instincts.
Without input there would be very little our instinctive 'lizard brain' could do. Our cerebral cortex wouldn't have a whole lot of information in it to do anything.
Think about this. If we take a newborn baby and restrict all sensory inputs (use your imagination because restricting all inputs would probably not be possible without essentially rendering it dead or a vegitable). This child can not formulate any new thoughts because it would have nothing to cause this to occur. No senses means nothing changes. Nothing to react to, nothing to think about.
If we assume however that we were given a certain amount of knowledge prior to cutting off our sensory inputs, it may be possible to conceptualize mathematics based on thought and contemplation over the existing intelligence we had already obtained. Simply thinking about our fingers and toes for example could spawn the beginning of mathmatics
I suppose if you take Plato's theory of Forms as the starting point, then you could say that abstract mathematical concepts are merely examples of Forms and therefore exist independent of our sensory experience.
On the other hand, the direct answer to your question ( are we "able to learn mathematics" if we had no senses) has to be "no." We are unable to accumulate any knowledge without senses. This isn't really a philosophical question as a scientific one - a human brain is a mesh of neuron connections that get formed over time based on input. Without input, neurons wouldn't train themselves to connect in new ways that lead to our accumulation of knowledge.
