To address your first paragraph.

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. If a primitive man stands atop a tower, looks around, and says, "The land is flat" (as far as he can see, it is), that is true, in that context. Sometimes you don't know the boundaries of your context, as in these simple cases. (I am well familiar with the problem of induction.)

But the thing is, with modern science, you can say "In the full context of modern thought, gravity behaves thusly," and the context is absolutely enormous.

You can't ask for more than that from induction. You can't ask for omniscience, and then, not getting it, throw the baby out with the bath water.

Note that there are no mathematical axioms here. I do assume that existence exists, that A is A, and that I am conscious. So those things have special status (for reasons I won't get into). But they are implicit in all claims to knowledge---even if you deny them, you are assuming them. They are not like mathematical axioms, which are just assumed and could be different (and are in different systems).

I don't know why you would say that deduction cannot be reconciled with induction. Let me give you an example.

Premise 1: All men are mortal.

Premise 2: Socrates is a man.

Thus: Socrates is mortal.

This is a deductive argument. However, how did we arrive at Premise 1? By induction. And it's true in a certain context. (And we all know what the context is; it not longer applies if Aubrey de Grey "solve death," for example, which is changing the context.) We also got Premise 2 by observation, which is kind of like induction, but is simpler.

Let me address your second paragraph.

I'm not making the error you suppose, which strikes me as a bizarre supposition, but that's because we are coming from such fundamentally different approaches. Anyway, I didn't say anthing about "true math" or "false math," that is your interpretation but it is not accurate.

What I am saying is that if I take one seed and add another seed and count them, I have two. In fact, if I have any number of seeds and add one and count them, I have one more than I did before. Moreover, if I plant five rows of seeds with five seeds in each row, it requires 25 seeds. Here, I am inductively discovering mathematical rules, instead of deducing them from axioms.

So you can have mathematical rules that are (or could be) gotten inductively from reality, and thus correspond to reality.

It would also be possible for mathematicians to come up with axioms that deductively lead to the same conclusions, and a lot of that goes on in mathematics. So if you start with Z-F set theory or Piano numbers or whatever (I'm not a mathematician), you can end up figuring out a lot of stuff (like calculus) that actually does correspond to reality. Though it's not coincidence that Newton did not discover calculus by reasoning from basic axioms, but did in trying to solve actual real-world problems using algebra.

Mathematicians also can and do come up with axioms that do not correspond with reality. That's fine, too, but it's not directly useful, though it can shed light on mathematics that DOES correspond to reality and be useful that way, and/or be used to develop techniques that are logical and then can be used to work with math that does correspond to reality, or whatever. So it's not totally pointless to do this. I'm not saying it shouldn't be done. I'm just saying, for example, that if I create a mathematical system where 1 + 1 = 1, or something can be true and false at the same time, that's fine, but it won't correspond to what we see in reality. I won't put two seeds into a pile and only have one seed, and I won't be able to have my cake and eat it too, even if that would be possible under a certain mathematical system.