This article states that there are more than just one form of logic:
http://aeon.co/magazine/world-views/logic-of-buddhist-philos...
http://aeon.co/magazine/world-views/logic-of-buddhist-philos...
All statements are true, false, arbitrary, or nonsensical.
Contrary to Priest, a statement can't be true and false. (It can be neither true nor false, by virtue of being arbitrary or nonsensical.)
A self-referential sentence, like "This sentence is false," is simply nonsensical. Famous philosophers have written whole books on the premise that this is an important problem, but it isn't.
Graham Priest would probably gesticulate that I just don't "get it," but no, I do get it. I've read the article, and it's bullshit.
In which category do you place the arithmetical statement that Godel showed how to construct?
It turns out that if you can prove something is both true and false, then EVERYTHING is both true and false. For example, let's say you want to prove X. Note that Y is true. But then you obtain a contradiction (since Y is false)! Hence X is true.
No, that's not what he did.
He constructed a statement that is true, but that can't be proved in a certain logical system. The statement was basically "This statement is unprovable with these axioms", which can either be proved - meaning you've proved something false, or can't be proved, meaning the statement is true, but is unprovable.
Note that this uses two notions of "true" - provable (can be derived from axioms), and actually true.
Even the second notion of truth actually just semantic consequence of the second-order theory of the naturals, which is a mathematical formal concept, not quite the same as actual (ontological) truth.
I thought Goedel Sentences are true in the sense of being provable within a "larger" system of axioms.
You're right that there are "larger" theories that can prove the consistency of PA, e.g. PA + existence of a large cardinal proves the consistency of PA. But, per Gödel, no consistent axiom system large enough to contain PA can prove its own consistency. This is probably a lot less bad than it sounds though; after all, if we have some unknown axiom system T, and we have a proof in T that T is consistent, does that really tell us anything? Because if T isn't consistent, then it can prove anything, including that T is consistent.
I'm missing something here. Is the statement "the three angles in a triangle add up to 180 degrees" true, false, arbitrary, or nonsensical? What do you call a statement that lacks (or reveals the lack of) crucial context?
That is absolute hokum. The Earth orbits the Sun, living things need fuel to survive, two units combined with two units sums to four units. I could go on and on.
You need good epistemology to explain why this is, since while it's common sense, I agree that common sense is not sufficient when we're talking about philosophy. Clearly you have not discovered good epistemology.
I took this to be similar to what Einstein [1] said: "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."
[1] http://www-groups.dcs.st-and.ac.uk/history/Extras/Einstein_g...
[1]: http://en.wikipedia.org/wiki/Gettier_problem
Also, see:
This is absolutely not my "position". The word know requires definition, which is a very hard thing to do, and is the subject of the whole of epistemology. All I'm saying is that "know" in the real world and "know" in logic are not -- and can never be -- the same. We cannot "know" something about the physical universe in the same sense that we "know" something to be true in mathematics (but we can certainly know a lot about the physical universe, if only we would define knowledge differently than in math, which is pretty much what every philosophy does). The point I was making is that you can't blindly overlay mathematical concepts on the real world merely because they use similar terminology. Particularly, the definition of "truth" is inevitably different in math and in science, if only because science does not have the concept of the axiom, which is a core principle in the definition of mathematical truth.
You cannot directly apply your notion of truth from logic to the physical universe and vice versa. Even in your examples you're confusing two different notions of truth. We do not know that the earth orbits the sun in the same sense that we know that 2 + 2 = 4. The latter is simply an algebraic axiom of the field of real numbers, while the former is an application of scientific assumptions to observations. We can be content with both types of knowledge, but we cannot equate them. We can treat both as true, but they are not true in the same sense.
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.
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.
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.
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.
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).
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.
You're too quick to reject it. The classic dichotomy works well in the objective sphere, but not so well on the boundary of objective and subjective.
There is awareness---I am conscious, so are you. It's a fact. It's true. There is simply no problem, here. It is not a great mystery. We don't know precisely how consciousness works, of course.
There is no boundary of the objective and subjective, because there is no such thing as subjective truth, which is an oxymoron. For example, if vanilla is my favorite ice cream, that is an objective truth about my personal preferences.
And which was that?
AI depends heavily on statements that are somewhat true and somewhat false at the same time.
How distorted does a letter shape have to be before it stops being readable? How much does readability depend on context? How much does meaning depend on context? What if input is inherently ambiguous and noisy but there's a signal under there somewhere?
As for smarts, there are different kinds:
Abstract symbolic reasoning (e.g. pure maths)
Using abstract symbolic reasoning to predict possible futures (physics, engineering, finance, AI)
A talent for optimising choices towards an explicit goal through abstract reasoning about possible futures (all of the above, with brain-powered decision making)
The same, but using unconscious processes ('intuition') that are not explicitly abstract or logical (all brain, no AI so far, but still effective)
The author seems to be suggesting that the first two kinds of smarts don't necessarily translate into the second two kinds, for various reasons. And not all smart people realise this.