First Brad Delong claimed that we can’t know things by pure reason. In response, I offered a counterexample:
The ratio of the circumference of a (euclidean) circle to its radius is greater than 6.28 but less than 6.29
Now Delong attempts to “refute” this counterexample by observing that it doesnt tell us anything about neutron stars (!!!). Leave aside the fact that it actually does tell us quite a bit about neutron stars (the circumference-to-radius ratio for a neutron star is not equal to 2π, but you’d still be hard pressed to compute its value if you didn’t know what π was). The larger point is that knowledge doesn’t have to be about neutron stars to be knowledge. It can be knowledge about, oh, say, euclidean circles.
Of course, as DeLong rightly observes, “we reason like jumped-up monkeys using error-prone Humean heuristics on brains evolved to improve our reproductive fitness”. And of course it is equally true that we perceive like jumped-up monkeys using error-prone sensory apparatus evolved to improve our reproductive fitness. Yet DeLong appears to acknowledge that our perceptions are sometimes informative. (I use the word “appears” because, true to form, DeLong prefers hissing and stamping his feet to actually spelling out an argument.) Why, then, should reason be more suspect than perception? DeLong isn’t in the mood to tell us.
This really should have been a comment over on DeLong’s blog, but I didn’t bother, because he’s notorious for deleting anything he can’t immediately ridicule — unless of course it supports him, in which case he’ll let the most egregious nonsense past. See, for example, the very first comment on the post in question, where mathematics is characterized as “the careful study of a set of tautologies” — which is true in (and only in) exactly the same sense that physics, psychology, and economics are careful studies of sets of tautologies. But mathematicians are generally more interested in figuring out which tautologies are true — just like physicists, psychologists, and economists this side of Berkeley.
Seems like an odd topic to get worked up about.
Shall we go for “how many angels can dance on the head of a pin” next?
JohnW:
Seems like an odd topic to get worked up about.
There’s some backstory here.
All tautologies are true. And scientists are not so interested in tautologies. They prefer to test hypotheses against real-world observations.
DeLong used neutron stars as his example because a black hole has infinite radius.
DeLong said that reason does not give us “direct access” to reality. I am not sure that is quite the same as we cannot know things by pure reason.
Nagels point – if the sun rises on the right and we believe we are driving south we “know” that our beliefs are in conflict. Therefore our reason has a direct link to reality. DeLongs point: It is possible that we are heading south and the sun rises on our right, therefore Nagels proposal that this demonstrates a direct link to reality is false. DeLong seems to be trivially correct- Nagel has left unstated certain asumptions which would make his example correct – i.e. we are not close to the pole, or on Venus, or some other solar system.
I don’t think DeLong was trying to say that we can’t know things by reason, just that our reason wasn’t “directly linked” to truth, as Nagel seems to claim.
Harold, I would agree with you if DeLong *merely* criticized Nagel’s particular exposition; you’re right, it was sloppy. But DeLong went on to say:
Thus Thomas Nagel’s insistence that we need a theory of consciousness that accounts for our reason’s ability to become an instrument of transcendence that grasps objective reality–that insistence falls apart like an undercooked blancmange…Any theory that provided such an account of reason becoming an instrument of transcendence and offering guarantees of grasping objective reality would be hopelessly, terribly, laughably wrong….
And I cannot help but think that only a philosophy professor would believe that our reason gives us direct access to reality. Physicists who encounter quantum mechanics think very differently…
I stopped reading DeLong a long time ago (no pun intended). I got the sense that he may sometimes have interesting things to say, but I kept getting distracted by a persistent thought: when are his eyes going to pop out?
Second the censorship issue.
“No true Scotsman” eh? I’m new to the term but I have a great deal of experience using this line of argument to diss music I don’t like. It doesn’t apply when you say “an object not defined in Euclidean space is not a Euclidean object”.
I don’t know what these special-case counterexamples are supposed to prove, but then again I don’t know what DeLong and Nagel are arguing about. What does “direct access to reality” mean and who cares? Why not keep it to probabilities?
Roger: You’re right; I typed “black hole” where I meant “neutron star”. This is fixed now; thanks.
All I have to add is that even near a neutron star pi is the ratio of circumference to diameter if the circle is small enough.
Of course, the idea of a “limit” requires the kind of reason DeLong seems to dispute.
I intended to comment on this earlier, but got distracted. In a discussion about the uncertainty principle for signals (which states that the product of the Gabor timewidth and bandwidth must be greater than or equal to 1/4pi), I once heard someone say that:
—
“Despite the name, there is nothing uncertain about the uncertainty principle.”
—
I like that.
Notice how much faster someone can play a jig on the high-frequency notes on a piano than they can on the low-frequency notes. This is an often-overlooked phenomenon called The Piano Postulate.
Am I right in saying the shape of a neutron star is undefined in Euclidean space? That’s what I gathered from the discussion.
Martin-2: Am I right in saying the shape of a neutron star is undefined in Euclidean space? Well, yes, in the same sense that the role in the orchestra of the color blue is undefined. Neutron stars don’t live in euclidean space any more than colors live in the orchestra.
I must be missing something, because I just don’t get this debate. We (humanity) learn about the world through experience, experiment and observation, and we can extend that empirical knowledge to other domains, including the domain of ideals, with mathematics and logic. Someone observed the path of a tethered goat and found the length of the path was about 6.29 times the length of the rope, and this was extended eventually to an abstract object we call a perfect circle, and pi was discovered. Why is this even controversial?
Neil (13): I take issue with the view that empirical knowledge must precede reasoning. Measurements at the time didn’t give anyone the third significant digit of circumference/radius. You might say nobody could have thought about circles so clearly without observing some physical ones, but if that counts then I can say people first had to reason they could learn about circles by marking a rope evenly and measuring them.
Steve (12): Actually, the role of the color blue is well defined in contemporary orchestras. OK I kid, but I was trying to ask if the sort of shape that describes neutron stars, with circumference / radius != 2*pi, is a Euclidean shape. I already know it’s not a Euclidean circle so maybe it was a silly question.
Martin-2
I chose 6.29 because that was the number Steve was using. I did not mean it literally. Presumably that early observer noted that the path was about 6 times the length of the tether, if even that exact. In any case, that eventually led to abstract reasoning that gave us geometric objects and their relations. Geometry, of course, means literally earth measurement, so its very name indicates that some sort of practical mundane and empirical practice preceded its abstract development.
Neil:
Someone observed the path of a tethered goat and found the length of the path was about 6.29 times the length of the rope, and this was extended eventually to an abstract object we call a perfect circle, and pi was discovered.
First, as Martin points out, no rope/goat combination ever gave anyone the idea that the ratio of the circumference to the rope is greater than 6.283185307179586476925 but less than 6.283185307179586476926. This has to have come from pure reason.
Second, I’ve used this example a bazillion times lately, but I keep coming back to it because I think it’s quite illustrative. Where is the goat/rope analogue for the fact that every natural number is the sum of four squares? This is not, as far as I can see, something that one is in any way led to through one’s senses.
“Where is the goat/rope analogue for the fact that every natural number is the sum of four squares? This is not, as far as I can see, something that one is in any way led to through one’s senses.”
No one is arguing that the senses themselves provide such knowledge directly. The proposition that every natural number is the sum of four squares comes from our ability to reason. But, reason is the process of properly integrating concepts, and it is impossible to create concepts without first having sense data to create concepts about. Therefore, all mathematical knowledge must eventually bottoms out in sense-experience.
To put it another way, the proposition that every natural number is the sum of 4 squares depends on first having the concept of numbers and rules about numbers. The concept of numbers depends on our early childhood experiences with perceiving quantities of objects. Those perceptions give us the resources to separate the concept of quantities from the actual objects themselves. For instance, just as we can separate the concept of “green” from green objects, so too can we use reason to separate the the concept of “5” from 5 objects. Likewise we can abstract rules based on our primitive observations and apply the rules to the concepts themselves. The iterative process of applying rules to more basic concepts will eventually reveal propositions like “all natural numbers are the sum of 4 squares.” Therefore it is only from that which is given, that we can construct what is not given.
So there probably wouldn’t be goat/rope analogue to the natural-numbers-are-the-sum-of-4-squares proposition except for the basic childhood experiences of observing quantities of objects.
I think mathematicians are more interested in beauty. The highest praise is “elegant” not “true.” Few believe both Tarski Banach and models where all sets of reals are true. They care that the web of implications are true, but not the resulting assertions. It suffices that A be true given B, and that A be beautiful.
Set of reals are measurable
I meant to say above
I need to learn to type
Let me expand on 18.
Mathematicians accept results whose truth they do not know in a sense. We all accept that TB follows from AC. We all accept that from other axioms all sets of reals are measurable. Despite their loose talk of existence mathematicians as a whole do not know wich theory is true or if true is even an applicable word here. But they think the proofs of each are valid and the results beautiful. Not because their transcendental truth detector bells rang.
Just as TB may or may not so may the relation of pi to the radius and circumference of a circle, and math provides no gyuidance about which you will find true. The qualification euclidean makes the truth analytic, not empirical.
I recommend Jamie Whyte’s comment.
I think that Steven is wrong to say that physics and mathematics are about tautologies in the same way. I assume we agree that (true) mathematical statements are always tautologies. For example, “the circumference of a Euclidean circle is 2*pi*r” is tautological. But it doesn’t tell us anything about the physical world. This is because whatever object we think may qualify as a Euclidean circle, may not! Whether it qualifies depends upon contingent facts. Similarly, “if a implies b, then ~b implies ~a” is obviously tautological, but again tells us nothing about the physical world. Perhaps we have mapped certain temporal patterns onto an implication relation – but we may be incorrect to do so.
The above point – that we cannot move from analytic to empirical facts – is the one Brad is making when he accuses Steven of making a ‘no true Scotsman’ argument. Just because we’ve constructed logic to work in a certain way doesn’t mean that our world has to work in that way also.
Let’s contrast the above with propositions from physics. For example, “the mass of the electron is 9e-31kg.” This is a contingent fact, not a tautology – there are possible universes in which this statement is false. Or consider Einstein’s field equations. These don’t have to be right a priori. They just happen to be right in our observable universe, as far as we can tell. And so on, with every fact about nature.
Notice also that my above comment is essentially in agreement with Ken’s #20.
A (somewhat) separate point: Brad’s argument, as I read it, is that sometimes people think that they are reasoning correctly when they are not. This is because we do not have ‘perfect access to empirical truths a priori’. If we want to learn about mathematics we can just sit in our armchairs, but if we want to learn about the world we need to go look at it. But this is a problem for Nagel – his argument depends upon us having a priori access to facts about the empirical world (which it isn’t clear that natural selection could give us, etc). And we don’t.
Gavin:
I think Nagel would agree with you.
“But reason depends on a constant supply of material from our pre-rational, animal nature—from perception, feeling, and natural desires. There is also an intermediate level of automatic judgment, some of it learned through experience, that operates more quickly than conscious reasoning and is essential for navigating the world in real time. The relation among these faculties is complicated. Even when we think we are using reason to arrive at the right answer to some factual or practical question—taking the relevant data consciously into account—our reasoning may be influenced more directly, without our knowledge, by the instinctive forces with which it coexists.”
http://www.tnr.com/article/books-and-arts/magazine/100050/reason-thinking-fast-slow-kahneman#
(For the record: no, I don’t read TNR, just stumbled across it.)
@Henri The link you posted shows that DeLong misunderstands Nagel.
Tyle:
Welcome to The Big Questions and thanks for your exceptionally thoughtful comments, though I disagree with much of this.
I. For example, “the circumference of a Euclidean circle is 2*pi*r” is tautological. But it doesn’t tell us anything about the physical world. This is because whatever object we think may qualify as a Euclidean circle, may not! Whether it qualifies depends upon contingent facts. Similarly, “if a implies b, then ~b implies ~a” is obviously tautological, but again tells us nothing about the physical world.
The statement “if a implies , then ~b implies ~a” is tautological because it remains true no matter what “a” and “b” mean. The statement that “the circumference of a Euclidean circle is 2 pi r” is NOT a tautology, because it becomes false, if, for example, you interpret “circumference” to mean “color”, “Euclidean circle” to mean “blad of grass” and “2 pi r” to mean “bright orange”.
II. Even if you use the word “tautology” loosely, so as to count “the circumference of a circle….” as a tautology (even though it’s technically not), it remains the case that math is not about tautologies. Take, for example, Mochizuki’s recently announced proof of the ABC conjecture (which I blogged about recently). Unless I’m mistaken, nobody has the foggiest idea what axioms he would need in order to formalize this proof — and nobody cares. They care that he’s discovered something *true*, not about the logical structure that’s needed to formalize it.
Likewise, if it were discovered tomorrow that the four-square theorem fails to follow from the Peano axioms (this is not going to happen, but consider the hypothetical), nobody would lose consequence in the four-square theorem; they’d just lose consequence in the Peano axioms. It’s *facts*, not tautologies, that mathematicians care about.
One more example: If you look at the Hercules/Hydra game that I wrote about in “The Big Questions” — we know that Hercules always beats the hydra, no matter how stupidly he plays — and we *know* that this can’t be proved in Peano arithmetic. But we still believe the theorem. I could give many more such examples.
III. Similarly, “if a implies b, then ~b implies ~a” is obviously tautological, but again tells us nothing about the physical world. So what? Since when is “knowledge” synonymous with “knowledge about the physical world”?
IV. You’ve claimed first that ““the circumference of a Euclidean circle is 2*pi*r” is tautological and then that tautologies tell us nothing about the physical world. But this fact about the circumference of a euclidean circle gives us *enormous* information about the physical world. without this — and the purely mathematical fact that pi is approximately 3.14 — you’d be hard pressed to calculate masses of elementary particles, for example.
V. A (somewhat) separate point: Brad’s argument, as I read it, is that sometimes people think that they are reasoning correctly when they are not. Both our reason and our senses sometimes fool us, ,so I don’t see how this can be dispositive for showing that our senses are necessary for knowledge and reason is not.
SL: “But mathematicians are generally more interested in figuring out which tautologies are true …”
You imply that some tautologies aren’t true?
no rope/goat combination ever gave anyone the idea that the ratio of the circumference to the rope is greater than 6.283185307179586476925 but less than 6.283185307179586476926. This has to have come from pure reason.
I suspect they also used quill and paper, or a sculpture carefully crafted out of copper and silicon, no?
Paul T: You imply that some tautologies aren’t true? Tautologies (like “if p then (p or q)”) are neither true nor false. *Interpretations* of tautologies (like “if grass is green then (grass is green or pigs are yellow)” are always true. But if you read my comment (and the comment I was replying to), you’ll see that I had agreed, for the purpose of this discussion, to use the word “tautology” in a broader and technically inaccurate sense — namely “things that can be validly deduced from a given set of (possibly false) axioms”, in which case, yes, of course they can be false.
The substance of this argument is one thing but a tenured professor and a colleague of Christy Romer posting titles like this: COMPLETE SELF-PWNAGE WEBLOGGING is downright kindergardenish. I have to wonder if people think twice before applying to the august abode where Dr. Delong resides.
SL: “Now Delong attempts to “refute” this counterexample by observing that it doesnt tell us anything about neutron stars (!!!).”
ho ho, this is doubly rich.
First, Mr. Delong claims that reason is an unreliable guide to reality, because he lost his bearings and got geographically confused once on an airplane. Well, he has successfully demonstrated one thing, in any case: UNreason can lead one to UNreality.
But the neutron star bit is icing on the cake. Let’s ask: how does anyone know they exist? First, physicists performed delicate experiments with exotic devices, to INDIRECTLY infer the existence of atoms, and their traits. Nobody has ever seen an atom.
Later, they extrapolated these theories to stellar physics, predicting the existence of neutron stars Finally, further observations INDIRECTLY provided confirmation.
“Indirectly”, because it was really an intricate chain of reasoning, which took centuries. The data, the ‘reality’, is merely patterns on film emulsion. Everything else is INFERRED.
Thus Mr. Delong references an immensely complex reasoning process, to show that reason cannot connect us to reality!
Credit where due, Mr. Delong is very talented, better than anything on teevee.
Steve @ 16 said:
“First, as Martin points out, no rope/goat combination ever gave anyone the idea that the ratio of the circumference to the rope is greater than 6.283185307179586476925 but less than 6.283185307179586476926. This has to have come from pure reason.”
Of course. That is what I said, or meant to say. We use our reason to extend and make precise the knowledge we derive from empirics. And you are correct–our observational powers may be in error as well as our reasoning powers, though I doubt I would make a mistake about whether the sun is rising on my right or left while I might easily unknowningly divide by zero. So the bottom line is we’ve got no disagreement. I’m not sure DeLong has any either.
I guess I was addressing your platonic claim that these mathematical truths exist somewhere outside our minds. Different topic. Sorry.
@Tyle: Yes, welcome, and thanks for nice comment. But one little trap on TBQ is that we are blessed with a plethora of Kens. There is Ken, and Ken B.
@Steve: We can prove the Hydra theorem from the normal axioms of set theory. It’s a consequence of any downaward sequence of ordinals being finite.
My examples of TB and measure are not tautologies either. They are true in some models, and false in others. I agree that mathematicians care about more than tautologies; the issue is why they care and what the evidence of their care and assent should mean to the rest of us. It should be taken as evidence of an insight into the nature of reality. The examples I just cited are contrary, they cannot both be true of the same space. Both are lovely, deep mathematics, and neither that I know of is disavowed by any mathematician.
Under IV, the postulate that the ratio is close to 3.141 … (800000 places), would be adequate for all the caluclations you note. These are physics calculations, the math makes them easier, not logically possible.
“It should NOT be taken as evidence of an insight into the nature of reality” I meant in 32
Sigh. Typing meltdown.
For those who did not study formal logic, here is a brief note on tautologies.
I will try to be clear about formal vs informal uses of some common terms, as this can cause confusion.
We have a language with rules for forming sentences. These will be quite strict mechanical rules.
We have rules for proving sentences from other sentences. These will be quite strict mechanical rules.
This we can say things like “From A, B and C we can prove D.”
As I said, we can test such claims of proofs mechanically, a computer could do it.
So proof can be defined quite formally and rigourously.
The notion of proof you will notice is unrealted to the notions of meaning and truth. It’s all about syntax.
We can model (using the word loosely) the idea of a sentence being true.
(I am simplifying here.)
The basic idea is this, we assign a truth value of T or F to each sentence, and we insist the proof rules
ensure that if A, B, C are true then so is D.
This is a restriction on the proof rules.
It is also a restriction on how we can assign T and F to sentences.
We can assign T or F in a lot of ways, but the restriction on the rules means we don’t have complete freedom. Some sentences it turns out must be assigned T for the scheme to work.
A tautology is a sentence (formal) that must be assigned truth value T.
An example is “Q or not Q”.
Only a very small amount of mathematics is a tautology.
Ken B, Steve is considering “0=1” a tautology because it could be derived from a faulty axiom such as “1=2”. Usually logic books say that tautologies are always true.
@Roger: I did say I was simplifying! But even then it wouldn’t be a taut because it can be F in some interpretations that fit the rules. A statement is a taut if it can only be given the value T, or the inference rules break.
To clarify 36, no Roger you are mistaken. Steve is using the standard and correct definition of tautology, which is a sentence whose truth value must be 1 in any model which respects the rules of logic. It’s a formal mathematical definition, but that’s the intuition of it: a sentence that must be true or you have contradictions.
Roger:
Steve is considering “0=1″ a tautology because it could be derived from a faulty axiom such as “1=2″.
You are mistaken; Ken B has this right.
If anyone cares, let me sharpen and summarized where I differ from Steve on some of these points.
This will make it easier to see where he errs, or if (horrors) I do.
Steve is basically right about what mathematicians do and how they do it.
He’s a bit off in an important way.
Mathematicians do not work from a set of axioms known to be true. Nor do they just play games with marks.
The ideas behind the marks are what they work with.
Axioms are chosen for usefulness, and to expose relations and structure.
(Noether’s restructuring of algebra shows this well.)
Neither do they just reject theorems if the proof requires a new axiom.
Quite the contrary, the need for a new axiom is often seen as a revealing an important discovery in itself.
Axioms often come late in the process.
Mathematicians prove results and judge them by their beauty and the soundness of their proofs.
They do not, as Steve implies judge, them by how true they are in ‘the real world.’
Accepting a new axiom is not a commitment to the truth of the axiom.
Mathematicians use ‘exist’ a lot.
But that can be seen as a short form for a long bit of folderol about provability, or as a kind of special ‘mathematical existence’.
Most mathematicans don’t worry (or even think) much about what the word might imply.
(@Steve: any disagreement so far?)
It’s wrong therefore to infer the corpus of mathematicians, attuned to truth, grasp that the number 4 “really exists” and might be spread upon toast and eaten (tastes like chicken).
Due to careless writing on his part, Delong is wrong on the narrow issue debated here.
He’s right on the broader issue I believe.
@Steve, your example of the four square theorem and the peano axioms brings to mind the attempt to prove whether parallel lines meet in euclidean vs non-euclidean geometry. I am no expert, but it seems that the axioms that a mathematician reasons from are more significant than you make them out to be. I am not sure how you can define true in a mathematical sense unless you talk about within a given set of axioms.
TravisAllison:
I am not sure how you can define true in a mathematical sense unless you talk about within a given set of axioms.
The four square theorem says that ‘every number is a sum of four squares’. Following Tarski, I define this statement to be “true” if, in fact, every number is a sum of four squares. Note that this definition makes no mention of axioms.
@TravisAllison: Axioms are used to define and classify what is provable within a theory. We have some choice in which axioms to use. They do not define what is true in the theory. We cannot choose what is true in a theory! Turh and provability are separate and distinct things.
This is not a controversial statement. There are numerous theorems establishing this.
So Steve is right on this point and DeLong is wrong.
The larger question is just what kind of knowledge do we get from pure reason, what is the relevance and applicability of these truths to debates about how the mind works. Some — Nagel — believe they tell us so much that they argue that minds cannot havce evolved in a darwinian way. That is the real claim DeLong is really rejecting, he’s just doing a bad job of rejecting it.
There is a wide range of opinion possible between DeLong and Nagel. I think Steve is a lot closer to Nagel than am I or is Keshav Srinivasan (who is the real expert on this topic on this blog), but he falls way short of denying evolution. In fact I think he probably claims his view supports the possibility of evolution better than ours does.
@KenB: thanks for the response. I am assuming that when you say that truth and provability are separate things, you are referring to the first Goedel Incompleteness theorem. But my understanding of the GIT is that there are ‘true’ statements that cannot be proved if the system is consistent. But the system cannot be proved to be consistent, if the system is really consistent. Would you say that the system is consistent is ‘true’? If someone disagrees that the system is consistent, there’s no method to show him why he is wrong.
So the idea that there are ‘true’ and unproven mathematical statements rests on the ‘truth’ of consistency. So to my way of thinking, ‘truth’ is rather empty. Provability within an axiomatic system and verifiability in general (physics, etc) is all that we can really have.
Obviously, this can get far into the weeds and I am sure that I’ve tried your patience enough with my amateur understanding of GIT. If you have a good book/article that you think shows that I am wrong, I’d appreciate a pointer. I’ve read Kleene, but if I remember correctly, I think he indirectly supports my position.
To the larger point of Nagel and his example, he seems to be saying that if you get a contradiction, you know that you’ve errored somewhere. I think that’s a pretty good heuristic, but I am not sure how you can arrive at this conclusion through pure reason rather than experience.
TravisAllison:
But my understanding of the GIT is that there are ‘true’ statements that cannot be proved if the system is consistent. But the system cannot be proved to be consistent, if the system is really consistent. Would you say that the system is consistent is ‘true’?
Just for concreteness, let’s take “the system” to mean first order Peano arithmetic. (If you don’t like that choice, feel free to make another.)
Then:
1) Surely, without knowing anything else about the subject, you can see that first order Peano arithmetic either is or is not consistent. Therefore one of the two statements “first order Peano arithmetic is consistent” and “first order Peano arithmetic is inconsistent” must be true, and would remain true whether or not it was provable.
2) First order Peano arithmetic is indeed provably consistent, though no proof of this fact can be formalized inside first order Peano arithmetic. Nevertheless, there are plenty of proofs, Gentzen’s for one — though my favorite proof is simply to note the existence of a model.
3) You can, of course, argue that any proof of consistency (or anything else) requires us to take *something* on faith. This is true, in exactly the same sense that any argument made in physics or psychology requires us to take something on faith — the existence of physical objects, or of other minds, for example. In this case, you’ve got to believe that simple combinatorial reasoning can’t lead you astray; you’ve got to believe that if (P and Q) is true, then P is true, and so on. You can choose to be the sort of extreme skeptic who doesn’t believe in anything at all, or you can choose to trust a few simple and uncontroversial principles. I recommend the latter path.
TravisAllison:
So the idea that there are ‘true’ and unproven mathematical statements rests on the ‘truth’ of consistency. So to my way of thinking, ‘truth’ is rather empty. Provability within an axiomatic system and verifiability in general (physics, etc) is all that we can really have.
Obviously, this can get far into the weeds and I am sure that I’ve tried your patience enough with my amateur understanding of GIT. If you have a good book/article that you think shows that I am wrong, I’d appreciate a pointer.
Torkel Franzen’s book is an excellent antidote to the sort of nonsense you’re espousing here.
TravisAllison:”So the idea that there are ‘true’ and unproven mathematical statements rests on the ‘truth’ of consistency. So to my way of thinking, ‘truth’ is rather empty. Provability within an axiomatic system and verifiability in general (physics, etc) is all that we can really have.”
Well I have not read Franzen but this really is wrong Travis. I really don’t think you can say or reason about anything if you don’t accept the very idea of consistency, or the consistency of predicate logic and the Peano Axioms. At least if you don’t I can just apply a ‘not’ in front of any of your conclusions, and do a victory dance. :)
I don’t spin out the metaphysical implications Steve does, but he’s entirely right in 44 [ except that using his ‘has a model’ proof is circular in this particular debate (which is about whether the model really exists)].
@Steve, thanks for reminding me about Gentzen. I’ve wanted to look at his result, but I haven’t because I thought I needed to build a better math base. I’ve read Franzen, but found it unsatisfying. He runs into the problem of using English for something that needs more precision. (Probably what he says makes a lot more sense if you already know the math behind what he’s saying.)
@KenB: “I really don’t think you can say or reason about anything if you don’t accept the very idea of consistency, or the consistency of predicate logic and the Peano Axioms.” Yeah, that brings us to the problem of what ‘to reason’ means. Would you say ‘to reason = to use a logic’?
Re 47: Includes and requires not equals.