During our brief intermission last week, commenters chose to revisit the question of whether arithmetic is invented or discovered—a topic we’d discussed here and here. This reminded me that I’ve been meaning to highlight an elementary error that comes up a lot in this kind of discussion.
It is frequently asserted that the facts of arithmetic are either “tautologous” or “true by definition” or “logical consequences of the axioms”. Those are three different assertions, and all of them are false. (This is not a controversial statement.)
The arguments made to support these assertions are not subtly flawed; they are overtly ludicrous. Almost always, they consist of “proof by example”, as in “1+1=2 is true by definition; therefore all the facts of arithmetic are true by definition”. Of course one expects to stumble across this sort of “reasoning” on the Internet, but it’s always jarring to see it coming from people who profess an interest in mathematical logic. (I will refrain from naming the worst offenders.)
So let’s consider a few facts of arithmetic:
- Every number is either odd or even. This is a tautology. It does not follow that every fact of arithmetic is a tautology.
- 1+1=2. This (depending on what you take as your starting point) is true by definition. It does not follow that every fact of arithmetic is true by definition.
- Every number is a product of prime numbers. This is neither a tautology nor a definition. It does, however, follow from the standard Peano axioms. It does not follow that every fact of arithmetic follows from the axioms.
- There is no solution to the equation (x+1)n+3+(y+1)n+3=(z+1)n+3. (This is the famous Fermat’s Last Theorem.) This is not a tautology, it is not a defintion, and I have no idea whether it follows from the axioms. Neither, as far as I know, does anyone else. All we know for sure is that it’s true.
- In Chapter 10 of The Big Questions, I give an example of a fact of arithmetic that is not a tautology, not a definition, and surely does not follow from the axioms. Here is a sketch of a proof that there must be some such facts.
It seems that people are often led astray by thinking that the “facts of arithmetic” consist solely of statements like “seven squared plus one equals five squared times two”. The more logically interesting statements are those that speak not about specific numbers, but about infinite collections of numbers. These are the statements that begin with phrases like “Every number…” or “There is a number such that…” or “There are only two numbers such that….”. When mathematicians speak of the “facts of arithmetic”, they means facts like these:
- Every number is the sum of four squares.
- Every prime number that leaves a remainder of 1 when divided by 4 is a sum of two squares.
- No prime number that leaves a remainder of 3 when divided by 4 is a sum of two squares.
- For every number n, there are infinitely many squares of the form 1+ny2
- Between every number and its double, there is at least one prime.
- 8 and 9 are the only successive numbers that are both powers of primes.
- For any two prime numbers p and q, the equations p=x2+yq and q=x2+yp are either both solvable or both unsolvable, unless p and q both leave remainders of 3 when divided by 4, in which case exactly one of them is solvable.
- If you want to black out enough squares on a tic-tac-toe board to make winning impossible, then, as you pass to boards of higher and higher dimensions, the fraction of squares you must black out gets arbitrarily close to 100%. This is the density Hales Jewett theorem, and I have intentionally glossed over some technicalities in the statement.
Now to the point: First, if you want to tell a story about whether arithmetic is invented or discovered—in other words, if you want to tell a story about where arithmetic comes from—then your story has to account for where the facts of arithmetic come from. If your story says that the facts of arithmetic consist entirely of tautologies, definitions and logical consequences of standard axioms, then your story is wrong. Try again.
I’m confused about the point that you “have no idea whether Fermat’s Last Theorem follows from the axioms”.
I thought that when a mathematician announces that they’ve “proven” a statement, they meant that the proof follows from some standard axioms and operations that are permitted under those axioms, and I had assumed those were the Peano axioms. (Even if such proofs don’t refer to the Peano axioms directly, I assumed they use higher-level manipulation steps which each encapsulate a set of operations permitted under the Peano axioms — like, adding a constant to both sides of an equation, without spelling out all the steps that let you do that.)
If that’s not what “proving” a statement means, then what does it mean? Did Andrew Wiles prove that Fermat’s Last Theorem follows from the Peano axioms plus some other axioms, like the axioms of geometry? (I know his proof used some geometry even though I don’t know anything about it.)
Bennett: Wiles’s proof uses set-theoretic constructions that virtually all mathematicians consider legitimate but whose legitimacy does not follow from the Peano axioms. Whether the proof can be rewritten to eliminate those constructions is (as far as I know) an open question.
As soon as you start talking about sets of sets of numbers, you’ve left the realm that the (first-order) Peano language can even talk about, let alone prove anything about.
No system of logic with a finite number of axioms can be both complete and consistent.
Or, equivalently, Any system of logic with a finite number of axioms is either incomplete or inconsistent.
Or, equivalently, any system of logic which is complete and consistent must have an infinite number of axioms.
Option three is where we are at. It isn’t that mathematical facts don’t follow from the axioms, it is that there are an infinite number of axioms required.
The principles used in Wiles’ proof may be derivable from simpler axioms, or not. If not, then they are, *quite obviously*, being adopted as axiomatic.
The attempt to spin some sort of mysticism or transcendence out of this is, in my opinion, wholly absurd.
Ben:
1) No system of logic with a finite number of axioms can be both complete and consistent. This statement is false, as are your equivalent variations.
2) It isn’t that mathematical facts don’t follow from the axioms, it is that there are an infinite number of axioms required. No set of axioms—finite or infinite—suffices to characterize arithmetic.
3) The attempt to spin some sort of mysticism or transcendence out of this is, in my opinion, wholly absurd. I have no idea what you’re talking about. The purpose of this post was to object to the drawing of metaphysical conclusions from false premises. Do you take issue with that purpose?
@Steve – Out of curiousity, what is your objection to Ben’s statement “No system of logic with a finite number of axioms can be both complete and consistent.” Godel’s first incompleteness theorem states “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.”
I assume what you’re objecting to is the distinction between “no system of logic”, because some simple logical system can be complete and consistent; and a system of logic that describes arithmetic, which cannot.
“Every number is either odd or even. This is a tautology. It does not follow that every fact of arithmetic is a tautology.”
Isn’t that statement false? I thought that parity was a property only of integers and that non-integers were neither odd nor even.
John Jenkins: I am using the word “number” here to mean “natural number”, which is the default interpretation when you’re talking about arithmetic.
Al V.: I assume what you’re objecting to is the distinction between “no system of logic”, because some simple logical system can be complete and consistent; and a system of logic that describes arithmetic, which cannot.
I (mildly) object to your emphasis on the word “simple”. The theory of euclidean geometry is both complete and consistent; I’m not sure whether you want to call it simple.
“Every short position has a corresponding long position”.
Watching the GS hearings yesterday I was baffled with the senators’ inability to grasp that it is impossible to generate a long position without an offsetting short position.
“Wiles’s proof uses set-theoretic constructions that virtually all mathematicians consider legitimate”
I did not know that proof in mathematics is a matter of consensus opinion. Do not those constructions need to be proved in order for Wile’s proof to be truly a proof?
Neil:
I did not know that proof in mathematics is a matter of consensus opinion. Do not those constructions need to be proved in order for Wile’s proof to be truly a proof?
The word “proof” means two different things. In mathematical logic, a proof is a list of statements, each of which is either an axiom or which follows from previous statements on the list. The “proofs” that are published in mathematical journals are nothing like that at all. They consist of informal arguments that reasonable people agree could probably be translated into proofs in the former sense, starting from some reasonable list of axioms, though in most cases it’s quite unclear exactly what those axioms are. In fact, there is an entire research area called “reverse mathematics”, where one takes established theorems and tries to figure out what axioms would be necessary to prove them formally; the fact that reverse mathematics is a difficult field shows that we don’t usually know what axioms we’re using when we prove theorems.
“No set of axioms—finite or infinite—suffices to characterize arithmetic.”
Can I take this to mean that even an infinite set of axioms does not suffice to characterize arithmetic? Pardon my lack of knowledge, but this does not seem right to me.
Joshua: Can I take this to mean that even an infinite set of axioms does not suffice to characterize arithmetic? Pardon my lack of knowledge, but this does not seem right to me. Yes, this is correct.
To be more precise, this is correct given the usual rules of the game, which is that you’re not allowed to adopt a set of axioms unless you have some procedure for determining what is or is not an axiom. You are therefore not allowed, for example, to say “I will take every true statement as an axiom”, because you have no procedure for recognizing which statements are true (and it is a theorem of Tarski that thre can be no such procedure).
You’re still allowed infinite sets of axioms; indeed, the usual axioms for arithmetic—the Peano axioms—are infinite in number. But they are all easily recognizable as axioms. On the other hand, they do not suffice to characterize arithmetic. There is much more on this topic in “The Big Questions” and in my earlier post on “non-simple arithmetic”.
“They consist of informal arguments that reasonable people agree could probably be translated into proofs in the former sense, starting from some reasonable list of axioms, though in most cases it’s quite unclear exactly what those axioms are.”
Interesting. Thank you. In my naivete, I would have thought that what you describe above is called a conjecture.
Here is another interesting fact: no one in the “intermission post” made a comment to the effect that mathematics is tautologous. However, I am happy that this can of worms has been reopened, because I subscribe to a modified form of that position.
Let’s have a look at two of the facts above:
There is no solution to the equation (x+1)n+3+(y+1)n+3=(z+1)n+3. (This is the famous Fermat’s Last Theorem.) This is not a tautology, it is not a definition, and I have no idea whether it follows from the axioms. Neither, as far as I know, does anyone else. All we know for sure is that it’s true.
Here we have a statement that is known to be true*, but we do not know whether it is deducible from the standard Peano axioms.
[* or is it? its truth depends on the self-consistency of the axioms it was deduced from, which is not guaranteed.]
In Chapter 10 of The Big Questions, I give an example of a fact of arithmetic that is not a tautology, not a definition, and surely does not follow from the axioms. Here [link to a post on Godel’s theorem] is a sketch of a proof that there must be some such facts.
Not sure about chapter 10, but the Godel statement is not known to be true: its truth depends on the self-consistency of the Peano axioms, which is not known as a fact.
So this is an example of a statement which is not deducible (nor refutable) from the standard axioms, but is NOT known to be true.
What I do not see is an example of a statement that is both known to be true and known not to be deducible from the standard Peano axioms.
Even if there were such a statement, in order to know that it is true we must deduce it from _something_, so even in this case truth (or more precisely, knowledge of truth) reduces to provability.
(I made this remark elsewhere, but having received no answer I feel entitled to repeat it.)
Snorri:
1) its truth depends on the self-consistency of the axioms it was deduced from No, no, no, no, no, no, no. It is true or it isn’t, quite independent of what axioms anyone chooses to write down. You might argue that our knowledge of its truth depends on properties of the axioms, but even so, it’s not the self-consistency of the axioms that’s critical; it’s their *truth*.
2) the Godel statement is not known to be true: its truth depends on the self-consistency of the Peano axioms, which is not known as a fact. No, the consistency of the Peano axioms (together with the axioms themselves) implies the truth of the Godel statement, but the converse is false. The Godel statement can perfectly well be true even if the Peano axioms are inconsistent. Note too that if they *are* inconsistent, then at least one of them is false, so there are plenty of other known theorems whose truth will also be in doubt.
3) So this is an example of a statement which is not deducible (nor refutable) from the standard axioms, but is NOT known to be true. This is correct in exactly the same sense that 2+2=4 is not known to be true. If you think the Godel statement might be false, then you must think that the standard axioms are inconsistent. If you think the standard axioms are inconsistent, you must believe that at least one of them is false. If you believe that at least one of them is false, then you can’t trust the proof that 2+2=4.
Doubting the truth of the Peano axioms seems to me to be on a par with doubting the existence of conscious minds other than your own. It’s quite impossible to refute such doubt, but also quite difficult to believe that you mean to be taken seriously.
4) What I do not see is an example of a statement that is both known to be true and known not to be deducible from the standard Peano axioms. If the Peano axioms are true, then they are consistent. If they are consistent then the statement in Chapter 10 of TBQ is true. (Ditto the Godel sentence.) That statement is therefore known to be true if the Peano axioms are known to be true.
5) Even if there were such a statement, in order to know that it is true we must deduce it from _something_, so even in this case truth (or more precisely, knowledge of truth) reduces to provability. And by this criterion, we can never know anything, because everything is deduced from something whose truth is in turn doubtable. Again—there is no way to refute this view, but I am extremely skeptical that you really believe that nothing is knowable.
Steven,
I wanted to comment on your genocide post, but the comments were closed out.
Still, I think this moral puzzle becomes more interesting when you consider the culture or society in question as an entity itself. Add all of the individuals in a society together and you have the society itself. In essence, sociology.
Now let’s take the “you” argument and suppose that on given day you loose a fingernail, chip a tooth, or some other minor loss. It may affect your behavior in the short term (nurse wound, limit physical exposure), but ultimately, it does not change the sum of the parts too much – you still procreate, contribute whatever it is you add or subtract to society etc…
However, when you are entirely eliminated, your impact is severely reduced and you cannot procreate, or add/subtract to society.
The first example could perhaps be a metaphor for what happens when you mass murder random people. The second could be a metaphor for genocide. As a historical example, the Romans made a huge impact upon western civilization and the premature elimination of their culture would have had devastating consequences for our western culture of today.
This can be related to the evolutionary chain blog post. Heads leading to intelligence etc… If the first species to have a head is wiped out, the consequence is dire for the development of intelligence. However, in evolution there have been many “mass murdering” of organisms within a species that could not adapt, but intelligence is still here regardless. Of course, this assumes that added complexity of life signifies progress and desirability.
Genocide may be akin to wiping out all organisms with heads, while random mass murders may be akin to natural selection. Which is worse?
Steve: I think it best to answer in reverse order:
“And by this criterion, we can never know anything, because everything is deduced from something whose truth is in turn doubtable.”
That describes my view (for which I claim no certainty, to avoid a contradiction).
The question is, why do you think this is problematic?
We act under uncertainty all the time anyway. Even if mathematics were known to be true with certainty (which would contradict Godel’s Theorem), that would not significantly reduce the uncertainty under which we operate.
Even when proving a theorem, the possibility of making a mistake is a much greater worry than the possibility that the Peano axioms are inconsistent.
“It’s quite impossible to refute such doubt, but also quite difficult to believe that you mean to be taken seriously.”
Here is the sleight of hand: we are supposed to accept the consistency of the Peano axioms based on commonsense; but having accepted it, we are supposed to believe that this commonsense view can be used to prove absolute truth.
This reminds me of the reasoning that led to the Challenger shuttle disaster: the mission cannot go ahead if the probability of failure is significant; therefore we assume that the probability of failure is negligible.
Part 2 of my answer — but first, I’d like to plea for a “preview” button for comment writers.
Now let’s deal with a few loose ends:
“If you think the Godel statement might be false, then you must think that the standard axioms are inconsistent.”
Correction: the standard axioms MIGHT BE inconsistent.
“the consistency of the Peano axioms implies the truth of the Godel statement, but the converse is false. The Godel statement can perfectly well be true even if the Peano axioms are inconsistent.”
This is beside the point: the point is that the Godel statement is not known to be true.
“You might argue that our knowledge of its truth depends on properties of the axioms, but even so, it’s not the self-consistency of the axioms that’s critical; it’s their *truth*.”
The only way in which it kind of makes sense to me to talk about the truth of *axioms*, is if you say that a set of axioms lacking self-consistency cannot be true.
Hence, it is self-consistency that is critical.
From my 1st comment: its [Fermat’s Last Theorem’s] truth depends on the self-consistency of the axioms it was deduced from
I should have written: the validity of the proof of its truth depends on the self-consistency of the axioms it was deduced from.
Snorri:
1) If your position is simply that you doubt everything, then I’m not sure why you’ve bothered (here and elsewhere) to address so many specific points. Once you’ve said you doubt everything, it adds little to say that you specifically doubt points A, B and C.
2) Even if mathematics were known to be true with certainty (which would contradict Godel’s Theorem) This suggests that you don’t understand Godel’s Theorem; I can’t imagine what contradiction you have in mind.
3) we are supposed to accept the consistency of the Peano axioms based on commonsense; but having accepted it, we are supposed to believe that this commonsense view can be used to prove absolute truth. No, you’ve got it backward. I’m not claiming that the Peano axioms are true because they are consistent; I’m claiming that they are consistent because they are true.
Happy to see that we are focusing on the main issues.
Steve: 1) If your position is simply that you doubt everything, then I’m not sure why you’ve bothered (here and elsewhere) to address so many specific points. Once you’ve said you doubt everything, it adds little to say that you specifically doubt points A, B and C.
There are degrees of doubt; the points that I am rallying against are points that I do not simply doubt: I am pretty confident that they are wrong.
Snorri: 2) Even if mathematics were known to be true with certainty (which would contradict Godel’s Theorem)
Steve: This suggests that you don’t understand Godel’s Theorem; I can’t imagine what contradiction you have in mind.
Godel’s SECOND Theorem (I forgot to specify it’s the second) proves that you cannot prove the self-consistency of an axiomatic system, except within a larger axiomatic system whose self-consistency is still in doubt. Hence the infinite regress, or recourse to commonsense.
Snorri: 3) we are supposed to accept the consistency of the Peano axioms based on commonsense; but having accepted it, we are supposed to believe that this commonsense view can be used to prove absolute truth.
Steve: No, you’ve got it backward. I’m not claiming that the Peano axioms are true because they are consistent; I’m claiming that they are consistent because they are true.
First, you say nothing about PROOF OF truth in your answer.
Second, I said nothing about truth OF AXIOMS in the statement that you quote. What I had in mind was truth of consistency, as well as the truth of the Godel statement.
However, I see that my quoted statement was in answer to a statement of yours about “doubting the truth of the Peano axioms”.
I should have made it clear that I do not doubt the truth of the Peano axioms, simply because I do not believe that the concept of truth can be sensibly applied to axioms. To me it makes no more sense to say that a set of axioms is self-consistent because true, than it makes to say that it is self-consistent because of uniform color.
Now if you believe that mathematics is discovered, then it might make sense to you to talk about truth of axioms; but short of circular reasoning you cannot then argue from the truth of axioms to the view that mathematics is discovered.
Snorri:
The only way in which it kind of makes sense to me to talk about the truth of *axioms*, is if you say that a set of axioms lacking self-consistency cannot be true.
This pinpoints at least some, and perhaps all, of what you’re not getting. Axioms are statements in the formal language of Peano arithmetic. There is a standard inductive definition (due to Tarski) of what it means such formal statements to be true. Truth is a property of a *single* formal statement; consistency is a property of a *collection* of formal statements. You are confusing two quite separate concepts.
Snorri:
To me it makes no more sense to say that a set of axioms is self-consistent because true, than it makes to say that it is self-consistent because of uniform color.
The difference is that a) there is a standard unambigous definition for the truth of a formal statement; there is no standard unambiguous definiion for the color of a formal statement. And b) it is a theorem that a collection of axioms, all of which are true, must be consistent.
@Steve Landsburg:”This statement is false, as are your equivalent variations”
Apologies, I did indeed mean a system of logic capable of describing arithemetic, although I suspect you knew that.
@Al V.
Thank you for the correction!
To rephrase: If the techniques used in Wiles’ proof are not proved to be derivable from axioms, then they are ipso facto being relied upon as axioms in their own right. There is no mystery about where they come from, they come by the choice of mathematicians.
So mathematics is partly invented (choice of axioms and techniques permitted) and partly discovered (consequences of those choices).
—-
I think your comment about the Peano axioms is probably closer to the nub of our disagreement.
You said ” I’m not claiming that the Peano axioms are true because they are consistent; I’m claiming that they are consistent because they are true.”
Why do you accept the Peano axioms as “true”? They just seem right.
Why do you accept Wiles’ techniques as valid? They just seem right.
I suggest that what you recognise as “truth” in the peano axioms, is that they and their immediate mathematical consequences have a close correspondence to the common sense instilled in you by evolution, which is there because they correspond well to the physical world. Other axioms you might like less, but they and their consequences might better model a different universe. This “truth” of the axiom itself is contingent, factual, empirical, in other words, and different in kind to mathematical truth.
Every truth of mathematics is either an axiom or a tautology. What it does not have is an independent existence, outside of those choices and those minds. That is what I mean when I say there is no “mysticism or trancendence” here.
Best regards,
Ben
Sorry about the repetition about the truth of axioms.
I kind of remember having read about Tarski’s definition of truth, but I do not recall that it can be applied to every statement in a formal language. Is a definition a statement? because a definition cannot be true or false, as your parents told you when introducing you to foreign languages.
Now if you view a set of axioms as a set of implicit definitions (though without a 1-to-1 correspondence between axioms and definitions), it follows that axioms cannot be true either. (Though you can say that they are false if inconsistent.)
Of course, if you postulate that mathematics is discovered, then axioms might not be implicit definitions.
Are you conceding the point that Godel’s 2nd theorem proves that there can be no absolute [i.e. independent of axiomatic system] proof of consistency?
Ben: Every truth of mathematics is either an axiom or a tautology. This is of course true, but only in the trivial sense that every *false* statement in mathematics is *also* an axiom; you are allowed to write down any axioms you like. But once you’ve committed yourself to a list of axioms (hopefully a list of true axioms), it is certainly not the case that every truth of mathematics is either one of *those* axioms or a tautology. In fact, it’s not even the case that every logical consequence of the axioms is either an axiom or a tautology. “At least one number is even” is neither a Peano axiom nor a tautology, even though it follows easily from the Peano axioms.
Snorri: I kind of remember having read about Tarski’s definition of truth, but I do not recall….
That would probably be a substantial barrier to our communication then.
In any event, your position seems to me to keep changing (which might just mean that I don’t understand it). Sometimes you seem to say that nothing is true; other times you seem to say that nothing can be known. It would clarify things a lot if you could tell me whether you agree with both, one or neither of the following statments:
A. ” `2+2=4` is not true. ”
B. ” We cannot know whether `2+2=4` is true.”
@Steve Landsburg: “”At least one number is even” is neither a Peano axiom nor a tautology, even though it follows easily from the Peano axioms.”
The statement “under Peano arithmetic at least one number is even” is a tautology.
Without an implicit understanding that we are talking about arithmetic, the statement would not be true, it would be meaningless. With such an understanding, as made explicit above, it is tautologous.
I am having trouble with your concept of a “true axiom”. Do you mean empirically true, given an assumed correspondence to physical phenomena? Or axiomatically true, assumed for the sake of argument? Or logically derivable from axioms?
These are three different things requiring careful distinction, but you are conflating them, linguistically if not mentally. I insist upon the distinction. If you do not agree they are distinct, we have successfully isolated our point of disagreement, and can stop here!
You said:
“But once you’ve committed yourself to a list of axioms (hopefully a list of true axioms), it is certainly not the case that every truth of mathematics is either one of *those* axioms or a tautology”
To make the distinction I insist upon, this should be one of,
Option 1:
“But once you’ve committed yourself to a list of axioms (hopefully a list of empirically true axioms (given an assumed physical correspondence)), it is certainly not the case that every *empirical* truth of mathematics is either one of *those* axioms or a tautology”.
That I agree with. Sometimes you have a logically undecidable proposition and you have to add an axiom. Hopefully you will pick one which matches your physical problem. This is the conventional conclusion of the Goedel’s paradox – we pick “true” because that corresponds most easily to our mental notion of truth. But that’s a choice.
Option 2:
“But once you’ve committed yourself to a list of axioms (hopefully a list of empirically true axioms (etc)), it is certainly not the case that every *logically derivable* truth of mathematics is either one of *those* axioms or a tautology”.
That I disagree with. The undecidable statements are neither *logically derivable* as truths nor as falsehoods. So they are in this sense neither axioms, nor tautologies, nor true, nor false.
They can still be true in the empirical sense (contingently), just not in the “logically derivable” sense.
Ben:
The statement “under Peano arithmetic at least one number is even” is a tautology.
You are very badly confused. The quoted statement is not even a statement in a formal language, so it’s not even a *candidate* for being a tautology.
I am having trouble with your concept of a “true axiom”. Do you mean empirically true, given an assumed correspondence to physical phenomena? Or axiomatically true, assumed for the sake of argument? Or logically derivable from axioms?
I mean true in the standard model.
“But once you’ve committed yourself to a list of axioms (hopefully a list of empirically true axioms (given an assumed physical correspondence)), it is certainly not the case that every *empirical* truth of mathematics is either one of *those* axioms or a tautology”.
What is an “empirical truth of mathematics”? I am not familiar with this concept.
“But once you’ve committed yourself to a list of axioms (hopefully a list of empirically true axioms (etc)), it is certainly not the case that every *logically derivable* truth of mathematics is either one of *those* axioms or a tautology”.
That I disagree with.
That’s because you’re badly confused. This is not a matter of opinion. From the Peano axioms and the standard rules of inference, I can derive the statement that there exists an even number, but that statement is not a tautology.
Sometimes you seem to say that nothing is true; other times you seem to say that nothing can be known. It would clarify things a lot if you could tell me whether you agree with both, one or neither of the following statments:
A. ” `2+2=4` is not true. ”
B. ” We cannot know [with absolute certainty, i.e. certainty independent of axiomatic system] whether `2+2=4` is true.”
I agree with B as I edited it, and strongly disagree with A.
Now it would help a lot if you told me where I “seem to say that nothing is true”.
Snorri:
I offered the sentence:
A. ” `2+2=4` is not true. ”
You said: I … strongly disagree with A.
But not long ago, you said:
I do not believe that the concept of truth can be sensibly applied to axioms.
Of course, `2+2=4` is a formal statement, and therefore can perfectly well be taken as an axiom. So by your quote directly above, it would seem that you do not believe the concept of truth can be sensibly applied to this statement. On the other hand, you strongly disagree with the assertion that the statement is not true.
This sort of gear shifting is why I suspect you haven’t thought very hard about what you’re saying.
`2+2=4` is a formal statement, and therefore can perfectly well be taken as an axiom. So by your quote directly above, it would seem that you do not believe the concept of truth can be sensibly applied to this statement.
No, it cannot be applied to this statement if it is taken as an axiom.
On the other hand, you strongly disagree with the assertion that the statement is not true.
… because I do not take it as an axiom.
PS: I asked where I seemed to say that NOTHING is true; not where I seemed to say that something is not true!
.
On the other hand, you strongly disagree with the assertion that the statement is not true.
… because I do not take it as an axiom.
Wait a minute. So we have a statement that is true. Then Snorri says “I take this statement as an axiom” and as a result, it stops being true?!?!??!?!
You are indeed attributing great powers to yourself.
This is really about trivialities:
Wait a minute. So we have a statement that is true. Then Snorri says “I take this statement as an axiom” and as a result, it stops being true
…for me: once I take 2+2=4 as the definition of “4”, it makes no sense anymore to say that 2+2=4 is “true” for me.
You are indeed attributing great powers to yourself.
That might be true in other contexts, but in this case my claim to power is no greater than the claim that I make for you; or for anybody with the gift of language, for that matter.
Anyway you still have not answered:
where did I imply that NOTHING is true?
Snorri:
where did I imply that NOTHING is true?
You said: I do not believe that the concept of truth can be sensibly applied to axioms.
Because every statement is an axiom in some theory, it follows that the concept of truth cannot be applied sensibly to any statement.
I realize you can’t possibly have meant this. I am pretty sure you didn’t mean anything else either.
2+2=4 is true because *whenever* we have a set where there we count two objects and we combine it with another set where we count two objects, and we find *invariably* that there are four objects in the combined set when we count them.
Should it ever be the case, and I do not lay awake at night worrying about this, that we find a case where we have a set where we count two objects and combine it with another set where we count two objects and, miraculously, count five objects in the combined set, 2+2=4 will cease to be true.
Snorri: I do not believe that the concept of truth can be sensibly applied to axioms.
Steve: Because every statement is an axiom in some theory, it follows that the concept of truth cannot be applied sensibly to any statement.
I see that you have not learned the lesson from your father. [I assume that you know which blog post I refer to.]
[a] A statement assumes a different meaning depending on the definition of the terms in it.
E.g. if I define 2:=1+1 and 4:=the number of planets in the solar system we are currently in [and note that this is not a constant number!] then it follows that 2+2=4 is not always true, and to the best of my knowledge it is not true for us right now.
[For the sake of simplicity I have retained the usual definitions of 1, number, planet, solar system, etc.]
[b] When a statement is itself a definition, then you could call it “true” in the sense that it is also a true theorem trivially derived from the definition itself. I grant you that. However, as a definition, it is neither true nor false.
[c] In any case, once you are committed to a set of axioms/definitions, then the truth of OTHER statements is no longer arbitrary.
Nor can you pick randomly any statement as an axiom/definition, if you want to retain consistency.
Therefore, my quote implies the opposite of what you claim it implies.
[d] If you are not convinced, then you should say explicitly that you do believe that definitions can be true or false, also giving examples of true and false definitions.
Snorri:
You keep saying thiings that are just plain false, such as this one:
Nor can you pick randomly any statement as an axiom/definition, if you want to retain consistency.
In fact every statement (excepting only negations of tautologies) is an axiom in some consistent theory.
More importantly, you seem to think that human actions (like “picking” axioms, whatever that means) can affect the truths of statements about arithmetic. I am, of course, using the word “think” rather loosely here.
And as for this: A statement assumes a different meaning depending on the definition of the terms in it., Right. Which is why I’ve told you 16 times that in this context, “true”, by convention, means “true in the standard model”, which is an unambiguous and well-defined criterion. You keep ignoring this.
Actually the “worst offender” was David Hilbert. So getting this stuff wrong is not “overtly ludicrous”. When I was a grad student I met math profs who knew nothing about this.
Otherwise a good discussion.
@Steve Landsburg:
I’m using “Tautologous” as meaning “logically valid”. I take it you are maintaining the distinction in terminology between facts which are tautologous in propositional logic and logically valid statements in first-order logic.
No-one would guess from your airy dismissal that this was a contentious distinction, argued over by Frege, Wittgenstien and Poincaré, and on both sides by Russell.
I hope that clears up the “confusion”.
You said originally:
“It is frequently asserted that the facts of arithmetic are either “tautologous” or “true by definition” or “logical consequences of the axioms”. Those are three different assertions, and all of them are false. (This is not a controversial statement.)”
Some facts of mathematics are true by definition, some are tautologous in the restricted sense you insist upon, some are logically valid. I don’t think anyone sensible argues that all facts of mathematics fall into the same one of those groups, though some argue that the distinction is unimportant.
The interesting set is those statements of mathematics which are none of the above. The Goedel statement being the canonical example.
The point I have been working towards, but not, it seems, successfully getting to, is that since such statements aren’t true in any of the above three senses (or possibly two, or one, if you don’t care about the distinction), in what sense are they true?
I am saying that there is, in addition, an empirical sense in which mathematical statements can be true or false.
They are true in the empirical sense, if, given an interpretation which relates propositions to real-world phenomena, these propositions express a concept which is empirically true.
And having, hopefully, now reached that point, I am saying it is in this sense that the Goedel statement is true.
Ben:
The point I have been working towards, but not, it seems, successfully getting to, is that since such statements aren’t true in any of the above three senses (or possibly two, or one, if you don’t care about the distinction), in what sense are they true?
They are true in the sense of Tarski’s inductive definition of truth. There is nothing “empirical” about this.
“They are true in the sense of Tarski’s inductive definition of truth. ”
That doesn’t help you. Tarski’s definition is itself based on a logical system which has to be either adopted axiomatically or proved, and is subject to the same problems of unproveable truths as arithmetic.
All you have done is peel the next layer. It’s elephants all the way down.
Ben: With regard to a certain equation F, Fermat’s Last Theorem (which, for all we know, is an example of a Godel sentence) is a formal version of the statement “Equation F has no solution”. This statement is true, by definition, if and only if there are no natural numbers that satisfy the equation F. Where’s the elephant?
I’ve told you 16 times that in this context, “true”, by convention, means “true in the standard model”, which is an unambiguous and well-defined criterion. You keep ignoring this.
No, YOU keep ignoring your own convention, which is why I said [without qualifications, i.e. taking standard conventions for granted] that I strongly disagree with ” `2+2=4` is not true. ”, while YOU keep saying that anything can be an axiom.
In addition to that, you keep quoting me out of context, which is why I am not going to clarify your latest befuddlement.
In addition to that, your use of the term “standard model” has been shifty in the past.
In addition to that, you have never mentioned the standard model in your replies to me in this thread.
Before going on, you would do well to address questions that you have been running away from:
A. can a definition be true or false? if so, please give examples.
B. can you adopt 4 = 2+2 as the definition of 4?
C. do you accept that Godel’s second theorem implies that any attempt to conclusively prove self-consistency of Peano arithmetic [or of any axiomatic system for that matter] leads to an infinite regress?
D. how can you KNOW mathematical truth, except by proof from axioms?
I’d like to call your attention to the fact that this is not the same as asking whether there are truths that cannot be proven from axioms.
Not that this should distract you from the questions, but I do believe that there is such a thing as truth.
Here is an example of a statement that I believe is true:
either Peano arithmetic is self-consistent, or it is not.
Obviously, I place more confidence in the truth of this statement than in the consistency of Peano arithmetic.
Here is a more general example:
Either a set of axioms is self-consistent, or it is not.
Either example, in conjunction with Godel’s second theorem, implies that there are truths that cannot be proven from axioms.
Neither of the examples, nor any implied unprovable truths, suggest to me that mathematics is about discovering some objective Platonic reality.
Snorri: The fact that you have to ask whether a definition can be true or false means that you are very much a novice at this stuff. That’s fine. We are all novices at most things. But it does suggest that maybe you’d do well to stop making bold and ludicrous claims about what you understand.
The answer to your question A is that the words “true” and “false” refer to the interpretations of statements within models. A definition is not an interpretation of a statement within a model. Therefore it is not true or false.
The answer to your question B is that you can adopt 4=2+2 (or for that matter 4=2+7 or anything else you like) as the definition of the symbol 4, but not, of course, in a context where you’ve already defined that symbol in some other way.
The answer to your question C is that I have no idea what you mean by “conclusively”. The Peano axioms are true in the standard model and therefore are self-consistent. You either do or do not find that argument convincing (I do), but I don’t see where infinite regress comes in. Alternatively, there is Gentzen’s proof of consistency, to which the same remarks apply.
Your question D is a non sequitur in the sense that proof from axioms is certainly not a guarantee of truth. Any formal statement can occur as an axiom and can therefore be proven from axioms.
If you have further questions, and if it appears that you are actually interested in understanding the answers, I will be happy to answer them, or to point you to sources where you can read more. On the other hand, if you are going to presume that everything you don’t understand is somehow my fault (e.g. when you don’t realize that invoking Tarski’s definition of truth *does* constitute an implicit mention of the standard model, or when you are unaware that “truth” in this context *always* refers to the standard model, or when your ignorance of the equivalence of various descriptions leads you to characterize those multiple descriptions as “shifty”) then I will absolutely ignore you.
The elephant is how do you know Tarski’s definition of truth is true? You can adopt it axiomatically, in which case you have a metamathematics in which you can construct Goedel sentences, proofs which are too complicated for us to be able to know whether they follow from the axioms, and all of the same problems.
I think we are talking slightly at cross purposes. Are you saying we know F is true by Tarski’s inductive definition of truth? How do we know that? I don’t think we do know that.
I think there are two problems here.
* What do we do about proofs which use techniques we can’t prove are valid? (The F problem)
* What do we do about statements which seem valid but we can prove that we can’t prove they are? (The G problem)
I’ve been more interested in the G problem, and you in the F problem, hence the cross purposes.
So for the F problem:
“(This is the famous Fermat’s Last Theorem.) This is not a tautology, it is not a defintion, and I have no idea whether it follows from the axioms. Neither, as far as I know, does anyone else. All we know for sure is that it’s true”
* We can I think agree that if it follows from the axioms then it is true.
* We can agree that it follows from accepting the techniques used in the proof.
So the problem becomes: What if the techniques used in the proof don’t follow from the axioms? Why would they be accepted, and why would we be satisfied that the proof is adequate if we can’t prove the techniques used are valid?
0. We could prove them.
1. We could give the proof techniques themselves the status of axioms.
2. We could accept them as true because they keep on producing satisfactory results, as compared to what we expect. I.e. option 1, but adding some shamefaced justification.
3. We could accept them on the basis that they seem intuitively true. I.e. still option 1, but without the shame.
4. We could accept them on the basis of some metamathematical system in which they can be proved valid, but which system has to be adopted axiomatically. I.e. Option 1, but with some intervening logical misdirection.
Now we are back on track!
Why did I not think before of ignoring quotes out of context?
The answer to your question A is that the words “true” and “false” refer to the interpretations of statements within models. A definition is not an interpretation of a statement within a model. Therefore it is not true or false.
This, I can agree on.
However, you do not seem to agree with yesterday’s Steve Landsburg who wrote:
there is a standard unambigous definition for the truth of a formal statement
Yesterday, truth was a property of statements, not of interpretations of statements.
In addition, axioms are not interpretations of statements, so I take it that today’s Steve Landsburg agrees with me that axioms cannot be true or false?
The answer to your question B is that you can adopt 4=2+2 (or for that matter 4=2+7 or anything else you like) as the definition of the symbol 4, but not, of course, in a context where you’ve already defined that symbol in some other way.
Thank you, this answer highlights the parallel between definitions and axioms, but if I had known your answer to A I would not have asked question B.
The answer to your question C is that I have no idea what you mean by “conclusively”.
Good point. I’d have clarified that, but I wanted to keep it short. However, I’ll clarify it below.
The Peano axioms are true in the standard model and therefore are self-consistent. You either do or do not find that argument convincing (I do)
I do too, with the qualification that it is _the interpretation of_ the axioms that is true in the standard model. See your answer A.
but I don’t see where infinite regress comes in.
It comes from the fact* that the standard model (in one of the more reasonable-sounding meanings of “standard model” that you used in the past) relies on axiomatic set theory, whose consistency can only be proven in a model which relies on some other set of axioms, whose consistency can only be proven …
So this is what “conclusively” meant in that context: “without relying on a set of axioms whose internal consistency is itself not conclusively proven”. This is a recursive definition of “conclusively” but I think we understand each other.
* Before you tell me that I have no idea what I am talking about, you’d do well to consider that, instead of “It comes from the fact that”, I could have written: “My understanding is that” or “Would you agree that”. But I seem to get more focused answers from you when I use the less tentative formulation.
Alternatively, there is Gentzen’s proof of consistency, to which the same remarks apply.
I am not familiar with that. However, let’s focus on Godel’s second theorem. (I trust that Gentzen did not prove Godel wrong.)
Your question D is a non sequitur in the sense that proof from axioms is certainly not a guarantee of truth. Any formal statement can occur as an axiom and can therefore be proven from axioms.
This is irrelevant to my question D as written, which I invite you to re-read.
Snorri: You continue to trumpet your ignorance and blame me for it. Truth is a property of interpretations of formal statements, not the formal statements themselves. However, it is also well-established standard practice to call a formal statement of Peano arithmetic “true” if its interpretation in the standard model is true. (This is exactly what Tarski’s definition formalizes.) Yes, the language is slightly sloppy, but it’s the language everyone uses. If the language, here or elsewhere, is unclear to you, feel free to continue asking for clarifications. But I won’t play your stupid “gotcha” game.
Incidentally, it’s a little hard to parse what you’ve written here but you appear to think that it is meaningful to ask whether a model is consistent. It isn’t.
Yes, the language is slightly sloppy, but it’s the language everyone uses.
Assuming for the sake of argument that “everyone” uses sloppy language, you are not obliged to follow suit.
If the language, here or elsewhere, is unclear to you, feel free to continue asking for clarifications.
When you say something that is just plain wrong if interpreted literally, “asking for clarifications” is a devious way of saying that you are wrong, which is what I have been saying. So you might as well say that I should be more devious.
Vice versa, if someone says that you are wrong (or asks you for “clarifications”) and you feel that you have been misunderstood, there is no point in blaming the sloppiness of the language “everyone” uses.
Now what about the truth of axioms? (NOT interpretation of axioms, just axioms. No sloppy language that “everyone” uses, here!)
And what about Godel’s 2nd theorem and infinite regress?
And what about my question D?
Snorri: Now what about the truth of axioms? (NOT interpretation of axioms, just axioms. No sloppy language that “everyone” uses, here!)
I have answered this already.
First: It is a red herring to ask about the truth of axioms. What you want to ask about is the truth of formal statements.
Second: Given an arbitrary formal language, there is no standard meaning for the word “true” as applied to a formal statement.
Third: Given the *particular* formal language of Peano arithmetic, the standard meaning of the word “true”, as applied to a formal statement, is “becomes true when interpreted in the standard model”. This is equivalent to “Tarski-true”.
Fourth: The use of the word “true” in the third point above is not the same as the use of the word “true” elsewhere in model theory. Like nearly every other word in mathematics (“normal”, “regular”, “smooth”, etc.) this word has multiple standard definitions that are understood to apply in various contexts.
Fifth: If you don’t like the standard language, that’s your problem, not mine.
Sixth: If you don’t know the standard language, that doesn’t mean the rest of the world is confused. It means you are ignorant (no shame in that!) and that you’d be better off asking questions than faking knowledge.
Definitional creep is a perrenial problem in philosophical discussions.
First you said (call it “the disputed assertion”):
“”It is frequently asserted that the facts of arithmetic are either “tautologous” or “true by definition” or “logical consequences of the axioms””.
Then, two days and 50 comments later, you explain that by “true” you don’t mean the millenium old, rather imprecise common usage, but an obscure jargon usage about half a century old from Model Theory, which will be familiar to less than 0.1% of the population, these people being “everyone” apparently.
Thank you for introducing me to that usage!
However saying “The arguments made to support these assertions are not subtly flawed; they are overtly ludicrous” cannot be justified based on the jargon sense of the word, since we can be sure that the people who make the disputed assertion are not using the word in that sense.
Arguing over the whole thread, on a blog intended, I think, to appeal to generalists, without once using the term “Model theory”, suggests that you are the one playing “gotcha”.
http://en.wikipedia.org/wiki/Tautology_(logic)
Third: Given the *particular* formal language of Peano arithmetic, the standard meaning of the word “true”, as applied to a formal statement, is “becomes true when interpreted in the standard model”. This is equivalent to “Tarski-true”.
Fair enough, but when I said that the truth of axioms makes no sense to me, you should have known that I did not follow this convention, instead of going off into an incoherent rant. Would you have been more coherent if I asked what you mean by “truth of axioms”?
Now let’s go back to definitions. Are they formal statements? because if they are, then it makes sense to talk about their truth as you define “truth”. (I am trying to explore the analogy between definitions and axioms.)
Fifth: If you don’t like the standard language, that’s your problem, not mine.
I like the language spoken by Torkel Franzen, Karlis Podnieks, and another whose name I can’t spell right now. I don’t like your language, when you talk about mathematics. (When you talk about economics, it’s a different story.) If your language is spoken by “everyone” except Franzen, Podnieks, and a few others, that only goes to show that I think more clearly than almost “everyone”. But then, almost everyone I actually know can think more clearly than almost “everyone”. I don’t see that as a problem.
I also like Roger Penrose’s language, though I would not say that he thinks clearly about Godel’s theorems.
Sixth: If you don’t know the standard language, that doesn’t mean the rest of the world is confused. It means you are ignorant (no shame in that!) and that you’d be better off asking questions than faking knowledge.
It isn’t faking knowledge to say that you are talking nonsense, and I already told you that asking questions, in this case, would be a devious way of saying that you are talking nonsense. One example:
Incidentally, it’s a little hard to parse what you’ve written here but you appear to think that it is meaningful to ask whether a model is consistent.
[a] I have never used the adjective “consistent” except in reference to sets of axioms;
[b] I almost never used the word “consistency” without adding “of a set of axioms” or “of the Peano axioms” or some words to such effect;
[c] when I did, “consistency” clearly referred to sets of axioms given the context.
Therefore, the assumption that I think it sensible to talk about “consistency of models” can only be made if you first assume that, like “everyone”, I use sloppy language.
You still have not addressed the issues of [C] infinite regress and of [D] how you can know mathematical truth.
Snorri:
(1) I like the language spoken by Torkel Franzen…I don’t like your language…
Perhaps you should go back and reread your Torkel Franzen, whose language does not differ from mine. (This is for the simple reason that both Torkel and I understand this subject.) You will find innumerable passages like the following:
“That the axiom `For every n, n+0=n` is true in PA means that for every natural number n, n+0=n. In this case, we know that the axiom is true.”
“We can always, when applying the incompleteness theorem to a system S, specify at least a subset of the sentences of S that express statements of arithmetic and are therefore either true or false. (In the case of S=PA, this subset is the entire system.)”
Et cetera.
(2) when I said that the truth of axioms makes no sense to me, you should have known that I did not follow this convention, instead of going off into an incoherent rant. Would you have been more coherent if I asked what you mean by “truth of axioms”?
You'll generally get more helpful responses if you don't start from the assumption that the various gaps in your knowledge are due to "shiftiness" on someone else's part, and if your "I don't understand" statements are not accompanied by intimations that I'm trying to pull the wool over your eyes, as opposed to trying to help you understand some very elementary mathematics.
(3) Now let’s go back to definitions. Are they formal statements? No. If you want to understand this stuff, you need to learn the definition of “statement”. Enderton’s book is a good place to start.
(4) It isn’t faking knowledge to say that you are talking nonsense Given that I am explaining elementary non-controversial material (in pretty much exactly the same words you could have found for yourself if you’d bothered to READ Torkel Franzen rather than just pulling his name out of the air), your bluster about “nonsense” is in fact pretty good evidence that you have no idea what you’re talking about and are trying to hide that fact. From whom, I have no idea.
(5) I answered your question about infinite regress already. Gentzen’s proof of consistency does not involve infinite regress. Neither does the proof that goes “The Peano Axioms are all true, and are therefore consistent.” Whether you find those proofs convincing is a judgment call.
(6) how you can know mathematical truth. I believe that mathematical truth, or at least arithmetical truth, is directly accessible to the intuition. Unlike the other stuff you’re trying to argue about, reasonable people might disagree with this.
Ben:
Then, two days and 50 comments later, you explain that by “true” you don’t mean the millenium old, rather imprecise common usage, but an obscure jargon usage about half a century old from Model Theory, which will be familiar to less than 0.1% of the population, these people being “everyone” apparently.
These are not different usages. The model theoretic definition is a precise statement of precisely the millenium old common usage.
Only in the same way that the botanical definition of a “nut” is a precise statement of the common usage.
For the common usage is not precise. If the model theoretic definition is precise they cannot possibly be the same.
Which is to say: “If it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.”
It might be true to say: “The model theoretic definition is intended as a precise statement of an important subset of the millenium old common usage, that can be rigorously reasoned about.”
People who say, like Russell in 1918, that all truths of mathematics are either axiomatic or tautologous, have generally never heard of the strict sense of “true” in model theory any more than Russell had, or the strict sense of “tautology” in first order logic. To know whether your criticism is valid you need to understand what they mean, not just how they spell it.
Otherwise you are acting like a person, who, offered a banana with the words “would you like some fruit?”, replies “What an idiot! A banana is a berry!”
Ben: I get your point, but I think it is misplaced. The precise model theoretic defintion says, for example, that the statement “For all x, P(x)” is true when and only when it is the case that for every number x, the statement P(x) is true. (And what does it mean for P(x) to be true? One proceeds inductively via similar steps.) This seems to me to be quite exactly what everyone has had in mind for millennia, not some important subset thereof.
Hi again.
Steve, your quotations from Franzen are out of context and therefore prove nothing about his sloppiness in the use of words.
That you use “slightly” sloppy language, is something that you have admitted yourself before I accused you of it — except for the specific case of “standard model”, which I said you use in a shifty way. However you admitted to using that term “slightly” ambiguously, in a previous thread.
That is not to mention your shifty use of the word “complexity” in that same old thread. It took me days to pin you down on that (in part due to your diversionary tactics) and then you complained that I am “extremely persistent and extremely dissatisfied”.
On a less personal note:
Snorri: It isn’t faking knowledge to say that you are talking nonsense
Steve: Given that I am explaining elementary non-controversial material […]
That you were not talking nonsense, is beside the point:
it isn’t faking knowledge to say that you are talking nonsense EVEN when you are NOT talking nonsense. To say that you are talking nonsense is [not always, but usually] to say that you contradict yourself, which can be detected [with false alarms, to be sure] without special knowledge; hence the accusation can be made without faking special knowledge.
Nothing personal in the following:
(5) I answered your question about infinite regress already.
OK, you said that proving Tarski-truth also proves consistency.
(I’ll leave aside Gentzen for now; if you think it appropriate to expand on it, please do so.)
There are a couple of problems with that:
First, a problem for your metaphysical position: you seem to have reduced truth to provability. You have not actually denied that there can be truths that are unknowable, but you have said that we know truth because we can prove it in the standard model.
Second, and most important for me, there is the issue that you seem to be doing everything to avoid dealing with: the standard model is based on axiomatic set theory. Any proof of “truth in the standard model” is based on the assumption of the self-consistency of axiomatic set theory. But how are you going to prove the self-consistency of the standard model?
I believe that mathematical truth, or at least arithmetical truth, is directly accessible to the intuition.
Sarcastic remarks come to mind, but I’ll restrain myself.
Except for a few random remarks:
I have been reading Ben’s comments more carefully and I wished I had done so before.
In one of his earlier comments, he distinguished [and I paraphrase him] between empirical truth and mathematical truth; with a further distinction between mathematical truth by assumption [axioms] and by logical proof [theorems].
However, that does not exhaust the list, because, as he recognizes, there are mathematical truths that are not provable. The example that I like best is the truth of statements of the form:
“Axiomatic system S is self-consistent”
Now that is not true for all values of S, but I certainly hope that it is true for some values.
Now you argue that such statements can in fact be proven, but you are using yet another sort of “truth”, namely, standard-model-truth: truth [which can be proven] in the standard model.
And at the end of your last reply to me you come up with yet another sort of “truth”, namely, Plato-truth, accessible only to the intuition.
That’s all: I do not draw conclusions for now. I want to go back and read Ben’s comments and your replies to him more carefully.
Erratum:
“But how are you going to prove the self-consistency of the standard model?”
Should have been:
“But how are you going to prove the self-consistency of axiomatic set theory?”
See why a Preview button is useful?
Snorri: Steve, your quotations from Franzen are out of context and therefore prove nothing
This, I think, removes any remaining shred of doubt about your seriousness. The quotations are taken from contexts where Franzen is carefully explaining what it means for a formal statement to be true, which is precisely what you arbitrarily and capriciously declared to be impossible. Your airy dismissal leaves open only two possibilities: Either you are an idiot, or you are feigning idiocy for sport. My money is on the latter. Here on the Internet, people like you are called trolls and we are admonished not to feed them. Having belatedly recalled that good advice, I am done wasting my time on you.
For the benefit of anyone else who might still be reading this, I should add that Snorri just made up that bit about the standard model being an artifact of axiomatic set theory, and that his question about the “consistency of the standard model” is laughably ignorant, since the word “consistency” applies only to theories, not to models. With any other poster, I’d have gently explained this. But I know from experience that if I tried to explain it to Snorri, he’d accuse me of lying, and if I pointed him to textbooks, he’d say they were “out of context”.
I think this is an interesting question, but I admit that I sort of zoned out on the debate. I wish someone could tell me, briefly and in plain English, exactly what the *substance* of the difference is between Steve and Snorri regarding why mathematics is or is not tautologous. (I already gather that Snorri thinks it is and Steve thinks it is not.)
Neil: Snorri’s position is that it both is and is not impossible for mathematical statements ever to be true and that he is prepared to defend both positions to the death, emphasizing at any given moment whichever one allows him to believe he’s being lied to. My position is that most facts of arithmetic are not tautologies, which is about as controversial as saying that most mammals are not bacteria. On the Internet, you can find someone prepared to “debate” just about anything.
@Steve Landsburg.
Yes, for mathematical truth I suppose you are correct and the definition is exact.
Thank you for taking the time to educate me – I have at least been persuaded that I need learn a lot more to have an argument worth refuting.
In the meantime may I congratulate you on your recent marriage, and wish you all the best. — Ben
Ben: Thanks for being here and for insisting that I do a better job. It’s a service to me and to the other readers. And thanks too for your kind good wishes.