> This is absolutely not my "position".

I didn't just say that to take a cheap whack at you. I'm not interested in that kind of low-brow discussion. But I see why you don't agree with that characterization. So I'll back off on that point.

I do think you're making a very serious error (albeit common in philosphers and mathematicians).

What you are saying is that mathematics is derived from a set of axioms, and therefore is true "by definition," whereas truth about the real world is something else. And you have a harder time in various ways) with non-mathematical truth. The foundations there are shakier.

However, the foundations actually are not shakier. They just aren't as widely known among philosophers (or common people). I'm not going to try to explain it here. That would be ridiculous. But to say one important thing that most people miss: You have to realize that knowledge is only valid in its context. It is contextual. So it is true to say, for example, that Earth orbits the Sun. It could turn out that we all live in the Matrix and that Earth doesn't orbit the Sun, but that claim would still have been true in the current context (where we have no reason to assume we live in the Matrix). Likewise, Newtonian mechanics can never be invalidated by later discoveries, because it is true in its context, i.e. at a macro scale. I don't know if this point will help you, but it is one barrier to getting scientific knowledge on a non-shaky foundation.

A separate point: while it's true that you can have a mathematical system with arbitrary axioms, you can also have a mathematical system that describes reality that is developed using philosophical (as opposed to mathematical) induction (just like all "scientific" truths), and thus that is actual true in a scienfic sense, and which is NOT based on axioms.

For instance, you can define a mathematics where 1 + 1 is 2, but you can also induce this rule from reality by looking at objects.

You can define a mathematics where 1 + 1 is 1, but you can't induce that from reality, because it isn't true in reality.

Really, the point of mathematics is to improve human life, not to be an idle pasttime for professors that don't have moral qualms with getting paid by taxpayers yet not rendering them a service. So the point of mathematics should be to look at mathematics as it can be induced from reality, and figure out things that are not only true in what you call a mathematical sense, but also true in reality. It's also fine to look at made-up mathematics, but only in that that sometimes turns into useful discoveries for real mathematics.

That anything can be induced from nature is a scientific assumption, and cannot be readily reconciled with logic. This is the "problem of induction". In any case, even if you decide to accept induction (and therefore causality) as "axiomatic truths of nature", even then you cannot reconcile mathematical deduction with scientific induction. Worse: induction is a constant "working assumption", as laws of nature "induced" from observation have often been shown to be false, or at the very least inaccurate, making even axiomatic induction (if you prescribe to that notion) a theoretical, yet unattainable ideal. This holds regardless of whether or not we live in the matrix.

I don't quite get your distinction between "real" and "made-up" mathematics, but I think you're affirming the consequent by presupposing a "true math", and then arguing that some set of axioms is "false" simply because it is not your own. In particular, the concept of truth and falsehood are not induced from nature because they are properties of statements about nature. Therefore, your own claim about the only possible truth values is not any more "real", or even induced from observation, than any other.

[1]: http://en.wikipedia.org/wiki/Problem_of_induction

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.

> 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.

To address your first point.

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.

Math is a philosophy.

The only difference is that Math adds a few extra requirements in that it is required to stay internally consistent using a given system of logic (in this case, strictly Boolean logic, it's the entire reason why proof by contradiction is even possible).

> Math is a philosophy.

No it's not. Not any more than chemistry is a philosophy or history is a philosophy.

> The only difference is that Math adds a few extra requirements in that it is required to stay internally consistent using a given system of logic (in this case, strictly Boolean logic, it's the entire reason why proof by contradiction is even possible).

Philosophy is also required to be consistent (properly, both internally and with the external world), and allows proof by contradiction without relying on strictly Boolean logic.

no.