Good read. I'm enjoying this.

But for all of your proposed beliefs, as to what a definition is, or what an assumption is, or a distinction, etc....  Where does this knowledge of yours come from?  How are you certain of it?  Not, how are you certain that "a bachelor is an unmarried man" (though I find the idea of that statement being inassailably true while we doubt the concept of men to be... questionable), but how do you know what definitions, themselves, are?  Or that things that are definitional are true?

And, relatedly, I've begun to wonder... we've been speaking about judgements from "sense data" as opposed to other sources (like presumably, where ever it is you feel you've received your knowledge about definitions and logic and so on), but how did you ever learn to seperate that stuff out from the judgements you've made on data from your eyes, ears, etc.?

How do you even know what "eyes" are?  Or that they provide sensory data?  What is sensory data, anyways, without relying on any judgements made based off of... well... sensory data?

How can you argue anything about judgements made based on sensory data (that they're accurate, inaccurate, arbitrary, or otherwise), unless you agree that sensory data exists?  And isn't "sensory data" meaningless, unless we believe that we've made accurate judgements to determine that there are things like eyes, ears, noses, and that we receive information from them?

The source of my definitions (where I got the idea of things to define) is of no consequence.  To ask if I know if the concept of definition is true is to already know it (like asking to define "define"; you already have to know it to ask it).  You can't question definition without using it.

I'm not quite sure I understand your second paragraph, but it looks like the next two are related, so I'll address those.

I do not know if I have eyse; I could be a brain floating in a vat recieving electrical stimuli, for all I know, or even less.  But, I do agree with the idea of the existence of sensory data; whatever "this" is, it can't be nothing (what else could I be referring to?).  It's what we do with it that I have issue with (making statements about other parts of reality, or proposing other parts of reality). 

 Well, I'm glad someone is!

So you have a problem with anything proved using mathematical induction, or its basic concept at least, as a basis?

I disagree.

I submit to you that, according to the views you've espoused, there's no way to distinguish between the judgements that you claim to be able to make with certainty (definitions, logic, and so on) and a judgement such as "the sky is blue."

Why can't definitions, logic, et al., come from the same place as my determination that the sky is blue?  Why is there any particular reason to parse it out into two different camps, one certain, the other nothing but assumption?

I'd also say that you can't question judgements on sensory data without making them.  It is via sensory data that we are aware that there is such a category.

No; math is (I think) a system of definition and logic (on the other hand, what was Russell trying to prove?).

I wish I could do your quotebox thing.

Anyways, if logic and definition were themselves questionable, then they wouldn't be questionable(and then would be, and so on).  They have to be true; that's how we can call anything true or false.  Perhaps this can be better answered with the next part here.

"I'd also say that you can't question judgements on sensory data without making them.  It is via sensory data that we are aware that there is such a category".

For one thing, this is already false; our judgements of sense data must be questionable because we know they sometimes are wrong (dreams, hallucinations).  The distinction here is that definition and logic are apart from sense data judgement, so it can be questioned with logic (well, everything is, but you see my point).  Nor am I questioning the idea that one can make judgements on sense data; it's just that they'll be arbitrary.

The concept is used outside of math as well, but math is the best way to explain it:

The basic premise of mathematical induction is that if you prove, or in this case accept, the base case as true you don't have to prove every case.  Instead of trying to prove every related case you simply prove that, in general terms, the next case is also true. 

A good example is helpful, but math is the best way to express it:

Problem: Prove that the sum of the first 'n' odd numbers is equal to n squared.

Let S(n) = the sum of the first n odd numbers greater than 0.

We need to show that S(n) = 1 + 3 + … + (2n – 1) = n² 
 
Base Case (n = 1):  S(1) = 1 = 1²

         The result holds for n = 1.
 
Induction Hypothesis:  Assume that S(k.) = k²
 
We must show that S(k+1) = (k + 1)²
 
S(k+1)  = 1 + 3 + … + (2k - 1) + (2(k + 1) - 1)

           = S(k.) + 2(k + 1) - 1      (by definition of S(n))

           = k2 + 2(k + 1) - 1      (by the induction hypothesis)
           = k2 + 2k + 1
           = (k + 1)(k + 1)      (factoring)
           = (k + 1)²
 
Therefore, we can conclude that since S(1) = 1 and that S(k.) logical implies S(k+1) then S(n) is
equal to the sum of the first n odd numbers for all n > 0.

The idea is that you can prove an infinite number of cases by proving a basic case and then proving a generalized case.  I think you can do something similar in the example I quoted as well.  Once you have decided to accept sensory data, even if reluctantly, it logically follows (logic being a result of our experience with our sensory input) that you can expound on this.  Even to the point of proposing other parts of reality.

You cannot expound upon sense data: we have no idea what it could represent (if anything) (or, so that's been my contention).  Yes, we can accept sense data, but not any judgements upon it (that is, deciding what, if anything is responible for sense data).