> First, you absolutely can do scientific induction. For example, if I drop a ball a bunch of times, I know that balls fall, as long as the context does not change.
You absolutely can do it, but that the result of scientific induction is meaningful is, itself -- at best -- a proposition that rests on scientific induction. Specifically, the proposition that "my memories of patterns of things which I have observed is a useful basis for predicting future observations from other acts or observations" is a proposition that, itself, must either is unsupported or is a generalization from past experience of exactly the type it supports.
> What I am saying is that if I take one seed and add another seed and count them, I have two.
You can't count them and get "one" or "two" until you define what one and two are, at which you have already made one plus one equals two deductively true, and the same is true of any other supposedly induced mathematical truths -- in order to be able to induce them, you must first have definitions, from which deductive mathematical truths follow of necessity.
Let me state your argument in the most general form: You need reason to validate reason.
That is true, but it's completely fine.
Ultimately you have to assume certain things. In Objectivism they are stated as: Identity (A is A), Existence (Existence exists), and Consciousness (And I know it). They are called "axioms," but they are not like mathematical axioms. Rather, they are rules that must be assumed in any claim to knowledge, including in any claim to deny them. For example, if identity is invalid, a claim like "Identity is invalid" would be meaningless.
Another difference from axioms as they are used by mathematicians is that nothing is deduced from these axioms; they are just the prerequisites for induction, any induction at all, from reality.
To address your second point.
First you need to induce the concept of "unit." Then, you can induce the concept "one," "two," and, if you want, other numbers.
Then you can come up with the concept of "addition." I'm not sure how you could deduce addition from the notion of counting numbers. If you'd like to explain it to me (without presuming it in the argument, or any other mathematical knowledge that we haven't gotten to yet), I'd like to see that. I'm skeptical that it can be done.
However, you certainly can inductively observe that groups of things denoted by counting numbers have a certain relationship, and then come up with addition that way, which is induction. And then you realize the causal reason for it being that way. But that is a normal thing in induction. Induction doesn't mean, "there is no cause for this to be true."
To summarize, my point is that the logic is something like observing "If I put one and one together, I have two," but that is induction, not deduction, prior to having addition defined (and then you do it deductively from then on because you already know the rule and you are just applying it).
Though at this point we only know about 1 and 1, we don't know about 2 and 1, for example. So to make it general, and have a general rule of addition, would require more inductive work.
This is all just me thinking through it. I don't claim to be representing Objectivism perfectly on this point, though I do on the first point.
You absolutely can do it, but that the result of scientific induction is meaningful is, itself -- at best -- a proposition that rests on scientific induction. Specifically, the proposition that "my memories of patterns of things which I have observed is a useful basis for predicting future observations from other acts or observations" is a proposition that, itself, must either is unsupported or is a generalization from past experience of exactly the type it supports.
> What I am saying is that if I take one seed and add another seed and count them, I have two.
You can't count them and get "one" or "two" until you define what one and two are, at which you have already made one plus one equals two deductively true, and the same is true of any other supposedly induced mathematical truths -- in order to be able to induce them, you must first have definitions, from which deductive mathematical truths follow of necessity.