So I say that truths of physics are not logically true or logically necessary in the second sense, but they, or some of them, are in the first sense--absolutely necessary, just as much and in the same way as 'All bachelors are unmarried' is absolutely necessary.

I doubt that it is necessary, though it may necessarily follow from certain other facts about nature, and if those are not absolutely necessary, then the law is hypothetically necessary.

Hypothetical necessity makes no sense. A proposition is either true on all possible worlds or it is not.

Hypothetically necessary truths are not true in all possible worlds, but are true in all possible worlds in which another truth is true.

For example, if it is necessarily true that minimum wage increases tend to bring about increases in unemployment, and if the enactment of a minimum wage increase is a merely contingent fact, then in all worlds in which the wage increase is enacted the tendency to increase unemployment will be found.

It is a commonly believed that only denials of truths of logic and math lead to contradictions that are absolutely impossible, because it is assumed that only these truths are (absolutely) necessary. However, Peikoff has already made a case against this, so now the burden of proof is on those who believe this to show that he is wrong. To assume this is to assume that only logic and mathematical attributes of things are necessary to them, while physical and other attributes are not. But why should we assume this?

Peikoff has made assertions. He has not made cases.

No, he has not just made assertions. He has argued for his claim. His arguments are contained in an article called "The Analytic-Synthetic Dichotomy". That is his case. We are here to attack or defend it, or attack part and defend part, and not just by assertions, but by arguments.

That fact that his position can be coherently argued against shows that he has demonstrated nothing logically necessary.

If "coherent" arguments are plausible-sounding arguments, then no. There are claims in mathematics whose truth or falsehood have not yet been ascertained. I believe that either Fermat's Last Theorem or Goldbach's Conjecture is one of these, and the other has been proved, but at one time neither had been proved. I assume plausible arguments could have been made for and against each. Yet eventually one of them was demonstrated to be true, and eventually the other may be demonstrated to be true or to be false.

Because we have not, nor can we, exhaust the physical cosmos to assert the contrary. All we can do is assume true what we have so-far observed. To assert, absolutely, the truth of some general statement that so far has been the case is to assert the impossibility of ever finding a counter example somewhere in the physical universe. Not only has this not been shown, it CAN'T be shown. We have not the time or energy to show it.

Again, you seem to be assuming something that needs to be proved: that all general statements about the world (=all general factual statements) are contingent. Prove it.

If any turn out to be necessary, and we can know these necessary truths, then we can now that they will hold in all cases.

It comes down to the old chestnut the induction is not a necessarily valid mode of inference. It is just a good guess and the kind of guess we have no choice but to make because there is no practical alternative.

That's not an old chestnut but a philosophical claim.

Assuming that you mean 'induction' in the modern sense and not in the traditional sense which Peikoff and Rand uses (and all philosophers before the late 19th century used), then yes, it is not deductively valid mode of inference, but that's merely because it is not deduction. It can still be inductively valid (though you may, with some logicians, prefer to avoid the terms 'invalid' and 'valid' when talking about induction instead use 'weak' and'strong'). Anyway, that is mainly a matter of terminology. It sounds like you do not agreewith falsificationists and Hume that induction is worthless.

Anyway, one of the few things Hume got right in this area is in seeing that the common belief that deductive connections are necessary and inductive ones absolutely contingent; rather, he saw that necessity is involved in both. So, if there are no necessary factual truths, as he thought, then we cannot do either induction or deduction, in factual matters. Fortunately, he was wrong.

Also, you are assuming that logical contradictions do not rest on facts, and that the facts about the actual world are never facts about all possible worlds (i.e., necessary facts), both of which need to be argued for.

Argument: Write a truth table. QED.

Are you saying that writing a truth table of the truth-functional statement will, if correctly done, prove the statement? I never denied that.

Perhaps you are be assuming that when I claim that such statements are factual that I am denying their necessary falsehood, because you assume that what is necessary (necessarily true or necessarily false) cannot be factual, and you assume that I share that assumption. I do not share it and challenge you to prove it.

The claim that necessary truths are not factual is the most important of all the claims we have been discussing that needs to be proved.

Given the semantics of logical connective and negations, anti-tautologies are in ALL CASES, false.

I never denied that, either.

Again, you seem to be assuming that when I claim that such statements are factual that I am denying their necessary falsehood because you assume that what is necessary (necessarily true or necessarily false) cannot be factual, and you assume that I share that assumption, which I do not.

A logical contradiction is not a particular statement of fact or otherwise.

Logical contradictions can be particular, but, more importantly, statements of fact can be either particular or general.

I think you main point is that logical contradictions are not statements about fact. I think you are assuming that logical contradictions are necessarily false and that necessarily false statements cannot be about facts, just as you probably assume that tautologies are necessarily true and that necessary truths are not factual.

But the claim that necessary truths are not factual needs to be proved--and it is the most important of all the claims we have been discussing that needs to be proved.

It is an instantiation of a general form p AND not-p. Always and forever false, given the meaning of AND and NOT.

I never denied that either. Again you seem to be assuming that in claiming that they are factual that I am denying their necessary falsehood because you assume that what is necessary (necessarily true or necessarily false) cannot be factual, and you assume that I share that assumption, which I don't.

Factually true particular (non-universally quantified) statements about the physical world may be true, if if true they so happen to be true.

If 'so happen to be true' means 'merely so happen to be true', then that is saying that they are always contingent, never necessary. But that needs to be argued for.

We know of no universally quantified statement about the physical world that is true.

Ba'al Chatzaf

I deny this. I say, for example, that "All water is H20" (that is, all water has the atomic structure expressed by that formula) is true and we know it to be true, because we know that being H20 is a necessary fact about water, not just a contingent one. So, to use the Twin Earth example from Putnam I discussed earlier, Twater is not water and cannot be water, because it is not H20.

Similarly, "All gold has the atomic number 7" is true and we know it to be true, because we know that being having that atomic number is a necessary fact about water, not just a contingent one. Now you may think otherwise, because you probably assume that the only attributes of gold that are necessary are the ones contained in the Nominal Definition---that is, the definition that we learn when we first learn the meaning of the term 'gold', such as "a shiny yellow metal" or something like that. However, by such a definition Fool's Gold would have to be counted as a kind of gold. But we never do: we always say that it is just something that is similar to gold.

Of course a non-analytic propostion would not yield a contradiction upon denial (though I deny that there are any such propositions), but how does that prove that non-defintional truths are non-analytic, rather than being a kind of analytic truths??

Simple: a non-definitional truth is by definition a synthetic truth.

If you want to define 'synthetic truth' this way, and define 'analytic truth' as definitional truth, I will accept that, for the sake of the argument, but then you must be accept the consequences of doing so: now you can no longer say that the following are true by definition:

Truths whose denial is a self-contradiction = analytic truths

Truths whose predicate is contained in their subject = analytic truths

because those presuppose

Truths whose denial is a self-contradiction = definitional truths

Truths whose predicate is contained in their subject = definitional truths

which needs to be argued for

(note: I do not deny that all of those the right of the = sign are included among those to the left of it, but I deny that all of those to the left of it are contained in those to the right of it).

You seem to have very circular reasoning--that is, you are assuming what you are trying to prove: I say you haven't proved that non-defintional are synthetic (i.e. not analytic), and your reply starts off by assuming that definitional truth are not analytic, the very thing I asked you to prove.

There is nothing to prove, it is a matter of definition (I suppose you meant "non-definitional truth").

See above.

Also, 'analytic' does not mean certain: if its denial is contradictory then it is analytic, whether we know it or not. (Remember, logical properties, such as being analytic, being self-contradictory and entailment, apply to propositons or relations between them, and they are independent of knowledge: for example, if p logicall entails q, that is true regardless of whether anyone knows it or not.)

The point is that anyone can derive the truth of an analytic statement from the statement and the given definitions alone and be certain about the outcome, as long as that person doesn't make logical errors. No empirical validation is needed. In that sense an analytical statement is certain, contrary to synthetic statements.

There can be uncertainties even about analytic truths, if we do not know how to prove them (for example, if Goldbach's conjecture and Fermat's last theorem are true, then they are analytic, but at least one of these remains unproven--I forget which; and many theorems in trigonometry were at one time not known.).

But I do take your point that they could be know with certainty if we found the proof.

But what you need to argue now is that no synthetic truth could be known in this way.

More importantly, that sentence was only an illustration. His general claims are:

(1) If a statement is true, then it is analytically true

A statement can be true while we don't know that it is true. In that case we can't logically derive the truth of that statement, and therefore it is not an analytic statement.

Again, a statement can be analytic even if we don't know it. For example, the propositions of trigonometry, being mathematical, are analytic and always were , even before anybody had deduced any of them from the properties of a triangle.

The propositions of trigonometry can be logically derived from the definitions. It is not relevant whether someone has or has not derived those propositions already. On the other hand a universally quantified statement (thanks to Bob for the terminology; this is what I meant, but I didn't know the correct term - I'll use this definition implicitly when I'm talking about synthetic statements) about the real world can never be logically derived from the definitions, as this would imply omniscience. That is the essential difference.

It is relevant that in some cases the propositions have not been derived yet, because it shows that the fact that they have not yet been derived does not show that they could not be derived, and so the fact that proofs of some admittedly factual propositions have not yet been so derived does not prove that they could not be.

And deriving universal generalizations about the world from definitions does not imply omniscience. It simply requires that they be necessary truths. And you will admit that necessary truths can be derived from definitions. What you will deny is that they are factual (about the world), because you think that necessary truths can't be about the real world. But you need to prove that.

These are some of the claims which Peikoff and I deny and which you presumably affirm (where "definitional truth" means a truth expressing a Nominal Definition, which is a definition we learn when we first learn the meaning of a term, and includes all truths of logic and all truths of math):

1. that only definitional truths are such that their denial yields a contradiction (i.e., are analytic)

2. that only definitional truths are necessary (i.e., such that it is impossible for them to be false)

3. that only definitional truths are non-falsifiable (i.e., certain, provable with certainty)

4. that definitional truths are not knowable empirically (i.e. from experience)

5. that definitional truths are "non-factual"--i.e., they "say nothing about the world"

Now you need to argue for each of the claims on this list without using any of the others claims on this list as a premise. Otherwise you will not make your case.

1. See above.

As I said above, you can define 'analytic truths' as definitional truths, and so you can prove the first part, but now you cannot define 'analytic truths' as 'truths whose denial yields a contradiction', and so you cannot assume that definitional truths are the only truths whose denial yields a contradiction, but rather need to argue for it.

2. The necessity of a synthetic truth has no meaning to me. Such a statement may be true in this universe but not in another one. What does "necessity" mean in that regard?

I don't know what your first sentence means---are you saying that you don't know what I mean by 'necessary synthetic truth' or are simply denying it? A necessary synthetic truth is one which is both necessary and synthetic: that is, it cannot be false (and so is true in every universe) but it is not definitional.

Now why do you think that it is impossible for a non-definitional truth to be necessary?

3. This is not coherent. Statements may be non-falsifiable without being true or certain. But only definitional truths are certain. See further above.

First, my original request was that you prove that only definitional truths are non-falsifiable, so it is only true statements that we are talking about at this point.

Second, you may say that a non-falsifiable statement may be uncertain, but clearly a falsifiable statement is uncertain.

Thirdly, you say that only definitional truths are certain--but that is part of 3, which what you are to argue for: simply asserting it is not enough. What makes you think that only definitional truths are certain?

4. This formulation is vague. The definitional truth may refer to something that is known empirically, but its truth cannot be derived empirically, as it cannot be falsified by empirical observation.

This assumes that all truths derivable empirically are falsifiable. That assumes the truth of 3 and 4:

3. that only definitional truths are non-falsifiable (i.e., certain, provable with certainty)

4. that definitional truths are not knowable empirically (i.e. from experience)

5. This formulation is also vague. A definitional truth may refer to things in the world, but it doesn't give any new information about the world, as its truth is independent of any empirical evidence.

The claim that definitional truths don't give any information about the world is precisely the claim that I want to argue for.

Yes, I do see that you said "new information about the world", and that is important. Yes, they don't tell you anything about the world that is not already contained in them, but that doesn't mean that they don't tell you anything about the world. And, yes, they don't tell you anything that you don't already know (unless it logically implied in and therefore deducible from what you know) when you learn the meaning of the word, but that does not mean that what you know is not factual.

For example, we can analyze the concept of triangles to come up with the basic truths of trigonometry, we can analyze the concept of bachelors to come up with statements such as ‘All bachelors are unmarried’, and we can analyze the concept of ice to come up with statements such as ‘All ice is solid’. Regarding the latter, you may not have certainty

Oh yes, you have..

The phrase "the latter" did not refer to any of those examples; as I remember, we were talking about Shallow and Deep Kinds, and those three were examples of statements about the former, whereas it is statements about the latter about which you may be uncertain.

, or any degree of knowledge, because you have to discover the basic truths: you have discover that water is H20, and that gold has atomic number 7,

I have serious doubts about that...

You have doubts about whether we had to discover those two facts? Surely you didn't mean that, or are you actually counting those as definitional truths?

Certainly not. Those truths follow logically from the axioms and definitions in mathematics, they cannot be falsified by empirical evidence.

I didn't say that they could be falsified by empirical evidence: I said that they were empirical--i.e., knowable from experience, without need for supplemental from any additional database or alleged database (such as innate ideas). Remember, I don't concede that all empirical statements must falsified; that remains on the list of claims for you to prove (see above).

What does "knowable from experience" mean? Do you think you can prove trigonometrical propositions by measuring enough triangles? Further I never said that empirical statements must be falsified, only that they must in principle be falsifiable, if they should have any meaning.

I think we encounter triangles in experience--that is, we encounter things that have triangle surfaces--and then from several such cases we abstract the concept of a triangle, and then we analyze this concept to derive trigonometrical propositions; so such propositions can be proved by analyzing the concept of a triangle, and that this concept is derived from experience. My point is that experience contains all of the information or data you need to ascertain the truth of trigonometrical propositions. Sure, this data must be processed by your rational faculty, but the data is still found in experience and so I say that we should not deny it the label 'empirical' simply because it had to be processed.

I meant to say above "I don't concede that all empirical statements must be falsifiable", and so that is what I want you to prove.

Peikoff never claimed to be authority (except on Ayn Rand). I would like to know where you and other people, such as Daniel, get the idea that is claiming to be.

Also, your claim that he is talking nonsense about science depends on your claim that he is wrong about the various dichotomies he has rejected, and you haven't yet done this (see above).

No, it doesn't at all depend on his being wrong on various dichotomies. See for example here and here. He even once said that he knew for philosophical reasons that the Big Bang theory had to be wrong, and that he was studying physics to be able to prove it. I kid you not!

I will have to look at those links.

I believe that Big Bang theory is probably true, because the great majority of scientists do, but I did hear one of who thought that the evidence for it was inadequate. It isn't irrational to doubt it.

I don't know Peikoff's reasons for doubting it. But if he was studying physics in order to prove it, and he came up with arguments based on physics, and not just philosophy to doubt it, you wouldn't object to that, would you.

I deny this. I say, for example, that "All water is H20" (that is, all water has the atomic structure expressed by that formula) is true and we know it to be true, because we know that being H20 is a necessary fact about water, not just a contingent one. So, to use the Twin Earth example from Putnam I discussed earlier, Twater is not water and cannot be water, because it is not H20.

Similarly, "All gold has the atomic number 7" is true and we know it to be true, because we know that being having that atomic number is a necessary fact about water, not just a contingent one. Now you may think otherwise, because you probably assume that the only attributes of gold that are necessary are the ones contained in the Nominal Definition---that is, the definition that we learn when we first learn the meaning of the term 'gold', such as "a shiny yellow metal" or something like that. However, by such a definition Fool's Gold would have to be counted as a kind of gold. But we never do: we always say that it is just something that is similar to gold.

It just so happens that a clear drinkable liquid necessary for sustaining life on this planet has the chemical makeup H20. On another planet in another galaxy with a completely different history of life evolution, this might not be the case. Who knows, somewhere some life forms might like liquid methane or liquid nitrogen which is clear. We don't know. But it is logically possible. The only living things we know about are water based. This need not be necessary. It could be that it so happens to be the case on this planet.

As to gold, it not only has to be shiny and yellowish but we also require that it be ductile and not oxidizable in air. Fools Gold is brittle and not ductile.

In any case we know of no synthetic general statement (universally quantifiable) that must be true in every possible world. So Kant's claim that such a statement exists is at worst false and at best an unsupported speculation. We know his prime candidate, the axioms of Euclid are not synthetic. They are conventional posits (i.e. just plain assumptions). We know that Euclidean geometry is not the only consistent geometry. At the time Kant wrote this, he did not know this to be the case. Kant also claimed Newtonian Physics was not only synthetic but necessarily true. We now know Newtonian physics is false, but close enough to being right to be useful.

As to mathematical truths, we prove statements of the following form: X follows from axioms A. That is an analytic statement based on the definition of "follows from". When we claim (for example) an statement in number theory is true, we are really claiming it either follows from the axioms of the real number system but limited to integers or it follows from the Peano axioms for integers. Some claims are conjectures. For example, the Goldbach conjecture which is not known to be true (i.e. follows from the axioms for integers or from the axioms for real numbers). The Fermat conjecture (miscalled Fermat's Last Theorem) was shown to be true by Wiles in 1993.

Kant's claim that apodictic synthetic statements exist is bogus. His entire analysis underlying -Critique of Pure Reason- comes crashing to the ground.

I deny this. I say, for example, that "All water is H20" (that is, all water has the atomic structure expressed by that formula) is true and we know it to be true, because we know that being H20 is a necessary fact about water, not just a contingent one.

'H2O' is a simple symbolic representation of water, water is not simply H2O. To say "All water is H20" is a gross simplification of the situation and should be re-worded to say "water is usually represented by the chemical formula H2O"

From wikipedia;

"Water can be described as a polar liquid that dissociates disproportionately into the hydronium ion (H3O+(aq)) and an associated hydroxide ion (OH-(aq)). Water is in dynamic equilibrium between the liquid, gas and solid states at standard temperature and pressure, and is the only pure substance found naturally on Earth to be so."

I deny this. I say, for example, that "All water is H20" (that is, all water has the atomic structure expressed by that formula) is true and we know it to be true, because we know that being H20 is a necessary fact about water, not just a contingent one.

'H2O' is a simple symbolic representation of water, water is not simply H2O. To say "All water is H20" is a gross simplification of the situation and should be re-worded to say "water is usually represented by the chemical formula H2O"

From wikipedia;

"Water can be described as a polar liquid that dissociates disproportionately into the hydronium ion (H3O+(aq)) and an associated hydroxide ion (OH-(aq)). Water is in dynamic equilibrium between the liquid, gas and solid states at standard temperature and pressure, and is the only pure substance found naturally on Earth to be so."

I did not say "Water is 'H20'". I said "Water is H20". I assume that, as a student of semantics, you know the significance of the inverted commas and so know the difference between H20 and 'H20', and the difference between water and 'water' (the use-mention distinction): 'water' is a term which refers to water, and 'H20' is a term which refers to H20.

In any case, to avoid any such confusion, I explicitly said what I meant by "All water is H20": namely, I said:

(that is, all water has the atomic structure expressed by that formula).

Now if it turns out that there the formula 'H20' can be used to refer to another chemical stuff as well as to water then you have a real criticism---though only if my example, and not of my point.

I deny this. I say, for example, that "All water is H20" (that is, all water has the atomic structure expressed by that formula) is true and we know it to be true, because we know that being H20 is a necessary fact about water, not just a contingent one. So, to use the Twin Earth example from Putnam I discussed earlier, Twater is not water and cannot be water, because it is not H20.

Similarly, "All gold has the atomic number 7" is true and we know it to be true, because we know that being having that atomic number is a necessary fact about water, not just a contingent one. Now you may think otherwise, because you probably assume that the only attributes of gold that are necessary are the ones contained in the Nominal Definition---that is, the definition that we learn when we first learn the meaning of the term 'gold', such as "a shiny yellow metal" or something like that. However, by such a definition Fool's Gold would have to be counted as a kind of gold. But we never do: we always say that it is just something that is similar to gold.

It just so happens that a clear drinkable liquid necessary for sustaining life on this planet has the chemical makeup H20. On another planet in another galaxy with a completely different history of life evolution, this might not be the case. Who knows, somewhere some life forms might like liquid methane or liquid nitrogen which is clear. We don't know. But it is logically possible. The only living things we know about are water based. This need not be necessary. It could be that it so happens to be the case on this planet.

You seem to be missing my point. Perhaps it is because you are unfamiliar with Putnam's Twin Earth example. I discussed it in this forum before but it was some time ago, so for you and anyone else who forgot it or missed it I will restate the main points:

Putnam asks us to imagine that we discover a planet very much like Earth, called "Twin Earth" On there is much that is like Earth (even people speaking a language similar to English, but I will leave out that part as I think I make my points without it). There is a material there which seems to be water. It is a liquid at normal temperatures and pressures, it is tasteless, colorless and odorless, it quenches thirst and it is found in their lakes and rivers (this list may not be exactly Putnam's, but it is close enough for our purposes): in short, it has all the superficial attributes of water). And we start calling it "water". Later, however, we do chemical tests on it and find out it that it is not H20. Putnam then poses the question: is it water, or not. That is, is it a new subkind of water, or just something that it is a lot like water?

The mainstream view, of the Logical Postivists and Ordinary Language philosophers, going back to Hume and still further back to Locke (and Descartes), too, commits them to the view that it is water. This is sometimes called the "description theory of meaning", which says that the meaning of the word is a certain description which we learn when we learn how to use the word, and so anything that fits that description is a referent of the term: in the case of water, whatever has the superficial attributes listed above would be water. As Locke would put it, whatever has the "nominal essence" of water (by which he meant the nominal definition) would be water.

However, Putnam denies that it is water. He says that it cannot be water because it does not have all of the essential attributes of water (by which he means the necessary attributes of water), because of the essence of what he calls "natural kinds" (which I call "Deep Kinds") has more to it than is contained in the description he gave. He says that because he says that the meaning of natural kind terms have an "indexical" (pointing) component: so, for example, it is part of the meaning of 'water' that it refer to stuff like this stuff (said while pointing to a "paradigm"--example--of water)--stuff having the attributes of this stuff, including attributes that may currently be unknown. (At this point is position is similar to Peikoff's: that a concept can include unknown attributes)>

As to gold, it not only has to be shiny and yellowish but we also require that it be ductile and not oxidizable in air. Fools Gold is brittle and not ductile.

Thank you for giving me an example to prove my case. You say that gold "has to be" not oxidizable by air and ductile, as well as shiny and yellowish. So you are conceding that those two attributes are necessary attributes, and yet those two attributes (or at least the first) are not part of a nominal definition of gold, but rather had to be discovered. So "All gold is not oxidizable by air" is an empirical and factual truth but also a necessary truth.

Now you may wish to say that you did not mean that to say of either "Gold is not oxidizable by air" or "Gold is ductile" that it is a necessary truth. However, if you deny that the latter is necessary, then you have no reason to say that Fool's Gold is not gold. How do you know that it is not gold?

In any case we know of no synthetic general statement (universally quantifiable) that must be true in every possible world.

I agree, if we use 'synthetic' as Kant did, to mean a truth whose predicate is not contained in its subject, or a truth whose denial is not a contradictory, because, as I have argued in earlier posts, there are no synthetic truths.

However, if you use 'synthetic' the way Cal does, to mean 'non-definitional' (where the definition is question is a nominal definition), then we do have an example of synthetic necessary truth, Ba'al": see the example above that Ba'al gave me.

We know his prime candidate, the axioms of Euclid are not synthetic.

Right, under either definition: math is analytic.

They are conventional posits (i.e. just plain assumptions).

As to assumptions being conventional, you are using 'conventional' in an odd sense. In any case, what are these assumptions supposed to be based on?

If we take 'conventional' in the usual sense then, as I have been saying, it is a mistake to think that definitional truths are merely conventional (it is because most of them are about Shallow Kinds that we think so, because Shallow Kinds have concepts so simple that we can learn them when we first learn the meaning of the word, and this misleads us into thinking that they conventional). However, if you disagree, then let's see your argument.

Kant also claimed Newtonian Physics was not only synthetic but necessarily true. We now know Newtonian physics is false, but close enough to being right to be useful.

I think in fact Newton's Three Axioms of motion are true and indeed necessarily true, because they follow from the concepts of force and body, just as trignometry follows from the concept of a triangle.

Yes, I know it is commonly thought that Relativity Theory refuted them, and indeed it seeems to have refuted much of the assumptions of Newtonian physics (though I came upon a post on the SOLO site, whose author seems to have been a scientist, arguing that this was a myth), but I don't think it refuted any of the 3 axioms.

What aspects of each of the 3 axioms do you think were refuted?

Note that one of the weirdier aspects of Relativity Theory, the claim that mass increases with acceleration, actually came about because of a refusal to abandon the 2nd axiom. (Although I think that this was not handled right)

One more related point: the Law of Addition of Velocities was not refuted, either.

As to mathematical truths, we prove statements of the following form: X follows from axioms A. That is an analytic statement based on the definition of "follows from". When we claim (for example) an statement in number theory is true, we are really claiming it either follows from the axioms of the real number system but limited to integers or it follows from the Peano axioms for integers.

To truly prove it the axiom itself must be proved, or else be self-evident.

Some claims are conjectures. For example, the Goldbach conjecture which is not known to be true (i.e. follows from the axioms for integers or from the axioms for real numbers). The Fermat conjecture (miscalled Fermat's Last Theorem) was shown to be true by Wiles in 1993.

None of which interferes with anything I said, and these examples in fact support one of my points, that to say that a truth is analytic or necessary is not say that it is certain: Fermat's conjecture is a mathematical truth, and so is necessary and analytic, but nonetheless was not certain or even known to be probable until 1993; the same is currently true of Goldbach's conjecture if is true.

Fermat's conjecture is a mathematical truth, and so is necessary and analytic, but nonetheless was not certain or even known to be probable until 1993.

Fermat's conjecture is a mathematical truth, and so is necessary and analytic, but nonetheless was not certain or even known to be probable until 1993.

Does anyone understand this??

Wiles proved Fermat's Last Theorem (so-called; it was really a conjecture) in 1993. Between the time Fermat stated it and the time it was proved, it was never proven conclusively but it was generally believed to be true. The attempts at proving Fermat's Last Theorem (so -called; abbr. FLT) turned out to be one of the most fruitful episodes in mathematics. Lots of new mathematics was developed just to prove FLT, for example, the theory of (algebraic) ideals invented by Kummer.

Just to make sure you know what is what FLT states there exists no integer larger than 2 for which the equation

At the moment I have no time for a long discussion, but I'd like to answer two points:

, or any degree of knowledge, because you have to discover the basic truths: you have discover that water is H20, and that gold has atomic number 7,

I have serious doubts about that...

You have doubts about whether we had to discover those two facts? Surely you didn't mean that, or are you actually counting those as definitional truths?

I have serious doubts about the statement that gold has atomic number 7, gold is rather heavy, you know...

As to mathematical truths, we prove statements of the following form: X follows from axioms A. That is an analytic statement based on the definition of "follows from". When we claim (for example) an statement in number theory is true, we are really claiming it either follows from the axioms of the real number system but limited to integers or it follows from the Peano axioms for integers.

To truly prove it the axiom itself must be proved, or else be self-evident.

Wrong. You shouldn't confuse a mathematical axiom with a Randian axiom, these are completely different animals. You cannot prove a mathematical axiom, that is exactly why it is an axiom! Neither is there anything self-evident in a mathematical axiom. The only thing you can say is that some axioms will generate more interesting theories than other ones, and for a meaningful theory a set of axioms shouldn't be self-contradictory, but that's all.

Fermat's conjecture is a mathematical truth, and so is necessary and analytic, but nonetheless was not certain or even known to be probable until 1993.

Does anyone understand this??

This is an example of a truth that is necessary and analytic, but was once not certain, which shows that necessity is not the same as certainty and analyticity is not the same as certainty.

What exactly do you fail to understand about that?

At the moment I have no time for a long discussion, but I'd like to answer two points:

, or any degree of knowledge, because you have to discover the basic truths: you have discover that water is H20, and that gold has atomic number 7,

I have serious doubts about that...

You have doubts about whether we had to discover those two facts? Surely you didn't mean that, or are you actually counting those as definitional truths?

I have serious doubts about the statement that gold has atomic number 7, gold is rather heavy, you know...

Yes. The atomic of gold is 79, rather than 7.

Which of course doesn't affect my point.

As to mathematical truths, we prove statements of the following form: X follows from axioms A. That is an analytic statement based on the definition of "follows from". When we claim (for example) an statement in number theory is true, we are really claiming it either follows from the axioms of the real number system but limited to integers or it follows from the Peano axioms for integers.

To truly prove it the axiom itself must be proved, or else be self-evident.

Wrong. You shouldn't confuse a mathematical axiom with a Randian axiom, these are completely different animals. You cannot prove a mathematical axiom, that is exactly why it is an axiom!

Your position is the Aristotelian position on self-evident truths, that they cannot be proved (though Aristotle believed they don't need to be) but this is one of the few cases in which modern logic actually is superior to Aristotelian: modern logic acknowledges that, trivially, every proposition follows from itself, and so can be deduced from itself, and this is true even of a self-evident propositions. So "All bachelors are unmarried" can be proven by being deduced from the self-evident truth "All bachelors are unmarried". Of course this is trivial and boring, and also superfluous, because self-evident truths are self-evident: once you understand their meaning, you see that they are true. So '2 + 1 = 3' can be known be true once one knows the meanings of '2', '1', '3' and of course '+' and "=" and understands the grammar of the sentence.

Neither is there anything self-evident in a mathematical axiom. The only thing you can say is that some axioms will generate more interesting theories than other ones, and for a meaningful theory a set of axioms shouldn't be self-contradictory, but that's all.

If mathematical axioms were neither provable nor self-evident they could not be used to prove anything true. You could only prove that a given mathematical proposition follows from the mathematical axiom, which may be what you mean when you say that you are proving the proposition. But if the mathematical axiom is not true and shown to be true then the alleged proof has given us no reason to think that the mathermatical proposition deduced from it is true.

So maybe you are saying that mathematical axioms are not true or false, or are not known to be. Is that your position?

Some time ago Daniel asked me to present my own summary of Peikoff's argument in the ASD. I was going to do so, but got busy and put it off.

However, then I got to thinking about it: this is a forum on the subject of Peikoff's ASD, and so people discussing it, or at least those who want to attack it, should have read it themselves.

Yet later I read someone say that Peikoff simply asserted his claims. So I decided to resume the project, or at least part of it.

Below is my paraphrase of what seem to be his main points. Later I may recast this in formal argument form, as I did several arguments of myself and others on this forum in earlier posts. But I decided to post what I have now. I have left out most of his summary of the views that he is attacking (though I may include this in a later post), since some have confused the views he is attacking (e.g., that a concept means only its definition) with his own views, and I have left out most of the other historical material.

Page references are to the ITOE, Expanded 2nd ed., 1990

Summary paraphrase of Peikoff's "The Analytic-Synthetic Dichotomy"

“Analytic” and “Synthetic” Truths

Analytic truths are those which can be shown to be true merely by analysis of the meaning of their constituent concepts. 94

A concept means the existents which it subsumes (the units which it integrattes), including all of their attributes 94(98)

Concepts mean existents, including their attributres, not arbitrarily selected subsets of these attribures 98

There is no basis for excluding some of the existents’ attributes from the concept’s meaning 98

Every truth about an existent X reduces, in its basic pattern, to:

“X [which means X, including all its attributes] is one or more of the things which X is.” 100

So when making a statement about X, our choices are to say:

“X is what it is” or

“X is not what it is”

So the choice is between tautology (in a sense) and self-contradiction. 100-1

So, in a sense, no truths are analytic, because no truth can be shown to be true merely by conceptual analysis, without prior discovery of the content of the concept (i.e., the attributes of the existents which the concept refers to);

in a another, sense, all truths are analytic, because the denial of a proposition ascribing an attribute to the existent contradicts the meaning of the concept (since the concept means the existent, including all of its attributes) . 101

‘I married a wonderful man’ does not mean ‘I married a wonderful combination of animality and rationality’. 104

Philosophers separate a thing from its characteristics, but, sensing a connection, say that the thing is one part of the meaning: that meanings have two components: extension and intension. 104

But this distinction is bad, because it separates an existent from its attributes. 105

Necessity and Contingency 106

The Law of Identity entails the Law of Causality, and so a thing cannot act contrary to its identity. Metaphysically, all facts are part of the identity of a thing, and so all metaphysical facts are necessary. Only facts resulting from human free-will choices are not necessary. Metaphysically, the Law of Identity determines how a thing will act, and it has no alternative to doing so. 109

Logic and Experience 112

Knowledge is not acquired by logic apart from experience or experience apart from logic, but by the application of logic to experience. 152

Three versions of this distinction are given below.

1. Logical v. Factual Truths

A dichotomy between truths of logic and truths of experience is analogous to dividing arithmetical truths into summational truths (truths which state the actual sum of a column of figures) and additive truths (truths arrived at by use of the laws of addition). 113-4

2. The Logically possible v. the Empirically Possible

The dichotomy between logical possibility and empirical possibility should be rejected because it requires assuming that a violation of the laws of nature would not involve a contradiction. But that cannot be, because a violation of the laws of nature would require an entity to act in contradiction to its own identity. 115

This dichotomy also presupposes the error of believing that a concept means only its definition.

This dichotomy is often argued for by saying that the empircally possible is conceivable while the logically possible is inconceivable 114-5

But ability to conceive otherwise does not alter the facts. 116

Furthermore, in the serious, epistemological sense of the word, you cannot conceive the opposite of that which you know to be true (apart from statements about man-made facts). If a proposition asserting a metaphysical fact has been demonstrated to be true then it has been shown that the fact is part of the identity of its subject, and so any alternative would require the existence of a contradiction. If a person knows this fact, then only by evading the fact could the person conceive the alternative. 116-117

3. The A Priori v. the A Posteriori

The view that some propositions are a priori---validated independently of experience, simply by analysis of defintions---is mistaken because definitions are based on experience. 117

The view that some propositions are a priori---validated independently of experience, simply by analysis of defintions---is mistaken because definitions are based on experience. 117

The statement "the sum of angles in a triangle in a Euclidean plan is 180 degrees" is true a priori. No one has ever experienced a triangle. There are no triangles in the physical world. Neither are there points, planes or lines. These things are purely abstract and live in human brains as concepts.

If mathematical axioms were neither provable nor self-evident they could not be used to prove anything true. You could only prove that a given mathematical proposition follows from the mathematical axiom, which may be what you mean when you say that you are proving the proposition. But if the mathematical axiom is not true and shown to be true then the alleged proof has given us no reason to think that the mathermatical proposition deduced from it is true.

So maybe you are saying that mathematical axioms are not true or false, or are not known to be. Is that your position?

A mathematical axiom is by definition true, therefore it is nonsense to talk about "proving" an axiom. A physical model may use some mathematical theory, but the question whether that specific theory (for example a geometric model) can be applied to a physical phenomenon is independent of the validity of that theory. You shouldn't confuse physics with mathematics. Roughly said, the analytic-synthetic dichotomy corresponds to the mathematics-physics dichotomy.

The view that some propositions are a priori---validated independently of experience, simply by analysis of defintions---is mistaken because definitions are based on experience. 117

The statement "the sum of angles in a triangle in a Euclidean plan is 180 degrees" is true a priori. No one has ever experienced a triangle. There are no triangles in the physical world. Neither are there points, planes or lines. These things are purely abstract and live in human brains as concepts.

Exactly. This is a crucial point that has to be understood.

If mathematical axioms were neither provable nor self-evident they could not be used to prove anything true. You could only prove that a given mathematical proposition follows from the mathematical axiom, which may be what you mean when you say that you are proving the proposition. But if the mathematical axiom is not true and shown to be true then the alleged proof has given us no reason to think that the mathermatical proposition deduced from it is true.

So maybe you are saying that mathematical axioms are not true or false, or are not known to be. Is that your position?

A mathematical axiom is by definition true, therefore it is nonsense to talk about "proving" an axiom. A physical model may use some mathematical theory, but the question whether that specific theory (for example a geometric model) can be applied to a physical phenomenon is independent of the validity of that theory. You shouldn't confuse physics with mathematics. Roughly said, the analytic-synthetic dichotomy corresponds to the mathematics-physics dichotomy.

I would have preferred to say a mathematical postulate is -assumed- to be true, for the sake of deriving its logical consequences. A mathematical postulate is unlike an -axiom- in the sense that Rand proposed. For example one can assume that more than one line can connect a pair of points. Obviously this is not the Euclidean line that we know and love. This is not the ideal of an infinitely thing thread stretched tight. By the way, straight line is derived from the Anglo-Saxon "strecht linen" which means a stretched linen thread.

That is why we can have Euclidean geometry and an infinite number of consistent non-Euclidean geometries. In point of fact if Euclidean Geometry is consistent then so is Riemannian Elliptical Geometry and Hyperbolic Geometry. That are either all consistent or all inconsistent (within themselves).

Whereas if one denies the axiom -- something exists--, one contradicts the denial since something has to exist to make the denial. That is the difference between an mathematical axiom (or postulate) and what Rand meant by an axiom.

The statement "the sum of angles in a triangle in a Euclidean plan is 180 degrees" is true a priori. No one has ever experienced a triangle. There are no triangles in the physical world. Neither are there points, planes or lines. These things are purely abstract and live in human brains as concepts.

We use undefined terms in every language, it is inescapable. In mathematics AND natural language 'point' would be considered undefined, and so one just 'knows' what it means. It is when we start defining words with other words that the difference between mathematics and natural language becomes evident. So if a circle is defined as the locus of points equidistant fro a point called the center we can see at one there is no such thing in nature.

If mathematical axioms were neither provable nor self-evident they could not be used to prove anything true. You could only prove that a given mathematical proposition follows from the mathematical axiom, which may be what you mean when you say that you are proving the proposition. But if the mathematical axiom is not true and shown to be true then the alleged proof has given us no reason to think that the mathermatical proposition deduced from it is true.

So maybe you are saying that mathematical axioms are not true or false, or are not known to be. Is that your position?

A mathematical axiom is by definition true, therefore it is nonsense to talk about "proving" an axiom. A physical model may use some mathematical theory, but the question whether that specific theory (for example a geometric model) can be applied to a physical phenomenon is independent of the validity of that theory. You shouldn't confuse physics with mathematics. Roughly said, the analytic-synthetic dichotomy corresponds to the mathematics-physics dichotomy.

I would have preferred to say a mathematical postulate is -assumed- to be true, for the sake of deriving its logical consequences. A mathematical postulate is unlike an -axiom- in the sense that Rand proposed. For example one can assume that more than one line can connect a pair of points. Obviously this is not the Euclidean line that we know and love. This is not the ideal of an infinitely thing thread stretched tight. By the way, straight line is derived from the Anglo-Saxon "strecht linen" which means a stretched linen thread.

That is why we can have Euclidean geometry and an infinite number of consistent non-Euclidean geometries. In point of fact if Euclidean Geometry is consistent then so is Riemannian Elliptical Geometry and Hyperbolic Geometry. That are either all consistent or all inconsistent (within themselves).

Whereas if one denies the axiom -- something exists--, one contradicts the denial since something has to exist to make the denial. That is the difference between an mathematical axiom (or postulate) and what Rand meant by an axiom.

Ba'al Chatzaf

Ba'al,

Yes. I agree with almost everything you say here. What Cal is talking about are postulates, not axioms (in Rand's sense or in any mathematical sense that I have ever heard of).

I like your point about the stretched linen thread. This should serve to remind us of the empirical origin of even our mathematical concepts.

Similarly, geometry began as more or less what we would call "surveying" today: 'geometry' literally mean 'measuring the earth'.

If mathematical axioms were neither provable nor self-evident they could not be used to prove anything true. You could only prove that a given mathematical proposition follows from the mathematical axiom, which may be what you mean when you say that you are proving the proposition. But if the mathematical axiom is not true and shown to be true then the alleged proof has given us no reason to think that the mathermatical proposition deduced from it is true.

So maybe you are saying that mathematical axioms are not true or false, or are not known to be. Is that your position?

A mathematical axiom is by definition true, therefore it is nonsense to talk about "proving" an axiom.

You cannot know that a statement is true unless it is either self-evident or provable by derivation (deductively or inductively) from the self-evident. Therefore you cannot know that an axiom is true unless it is self-evident or provable. Therefore you cannot know that it is an axiom unless it is self-evident or provable (note that I am not denying that it will still be axiom: statements can be true without us knowning them to be true).

I think you have axioms confused with postulates (see my last reply to Ba'al). You may assume a postulate p to be true in order see its consequences, or even to prove it false. Then if you can validly deduce q from p you have proved the truth of the conditional statement 'If p then q'. But you will not have proven q, if you have not proven p.

If q is false, and is either self-evidently false or provable as false by deduction from a self-evidently false statement, then you can prove p false (this is a "Reduction ad Absurdum" of p), though proving q true will not prove p true.

A physical model may use some mathematical theory, but the question whether that specific theory (for example a geometric model) can be applied to a physical phenomenon is independent of the validity of that theory. You shouldn't confuse physics with mathematics. Roughly said, the analytic-synthetic dichotomy corresponds to the mathematics-physics dichotomy.

Ttruths of physics and truths of mathematics differ in their subject matter, but not in their truth.

I know that the analytic-synthetic dichotomy is supposed to correspond to the mathematic-physics dichotomy, but I deny this--and do so even on your definition of 'analytic truth'--i.e., definitional truth--because I say that Newton's axioms can be derived from definitions of the terms 'body' and 'force', and so, even by your definition, are still analytic.

The view that some propositions are a priori---validated independently of experience, simply by analysis of defintions---is mistaken because definitions are based on experience. 117

The statement "the sum of angles in a triangle in a Euclidean plan is 180 degrees" is true a priori. No one has ever experienced a triangle. There are no triangles in the physical world. Neither are there points, planes or lines. These things are purely abstract and live in human brains as concepts.

Ba'al Chatzaf

As I said in early post, abstraction is a mental process by which selectively focus on certain attributes of something (and so ignore the rest), and the word 'abstraction' can also be used to refer to the result of that process, which is itself a mental entity, a thought or idea. It is common mistake to reify this and so suppose the existence of abstract entities outside of the mind, as Plato did.

So insofar as their it is true that these beings do not exist in the physical world we should say that they don't exist at all, but we have thoughts of them--just as we should say, when we trying to precies that vampires and Sherlock Holmes do not exist at all, but we have thoughts of them (as opposed to saying "Vampires and Sherlock Holmes exist in the mind", which is only true metaphorically: they don't literally exist even in the mind, but our thoughts of them do exist there).

Much more importantly, the fact that there are no perfect triangles in the world does not make that statement true a priori, because our ideas of triangles were derived from experience, even though indirectly: that is, we encountered approximate triangles in experience and acquired the concept of approximate triangles and then idealized the concept to form the concept of perfect triangles. But the material for this concept came from experience, and so it is an empirical concept. (Similarly with your example a perfectly straight line.) And the above statement was derived from this empirical concept. There fore it should be considered to be known by empirical (a posteriori) rather than a priori means.

(Actually, in the case of triangles and other geometrical beings the concept is even more closely tied to reality: even though there are no perfectly triangular surfaces, there are surfaces that look perfectly triangular to the unaided eye, and from these we formed the concept of perfect triangles, without even needing to idealize, and then only later, when we measured carefully, did we find that the triangles were not composed of perfectly straight lines.)

Here is a simple refutation of your claim: if the fact that there are no perfect triangles in the physical world proved that truths about triangles are not known empirically, then the fact that there are in the physical world no bodies that are not under the influence of any force would prove that Newton's First Axiom is not known empirically, but the latter claim is false, and therefore so is the former.

We use undefined terms in every language, it is inescapable.

Not if we take 'definition' is the broadest sense of the term, to apply to anything that indicates the meaning of a term. So a definition does not have to be verbal: there are ostensive definitions, which involve pointing.

In mathematics AND natural language 'point' would be considered undefined, and so one just 'knows' what it means.

How does know what they mean, without a definition?

Both could be defined, and in math terms normally are defined, and often need to be, because mathematicians often define terms differently from non-mathematicians (which is a very unfortunate habit they share with most other people: when people come up with a new concept they should come up with a new term for it--either a new word or a new combination of old words--but people are frequently lazy about that and grab a term that already has a meaning).

It is when we start defining words with other words that the difference between mathematics and natural language becomes evident.

Most words in natural language a defined, and defined verbally.

So if a circle is defined as the locus of points equidistant fro a point called the center we can see at one there is no such thing in nature.

See my last reply to Ba'al, concering the apriori.

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## Gregory Browne

Hypothetically necessary truths are not true in all possible worlds, but are true in all possible worlds in which another truth is true.

For example, if it is necessarily true that minimum wage increases tend to bring about increases in unemployment, and if the enactment of a minimum wage increase is a merely contingent fact, then in all worlds in which the wage increase is enacted the tendency to increase unemployment will be found.

No, he has not just made assertions. He has argued for his claim. His arguments are contained in an article called "The Analytic-Synthetic Dichotomy". That is his case. We are here to attack or defend it, or attack part and defend part, and not just by assertions, but by arguments.

If "coherent" arguments are plausible-sounding arguments, then no. There are claims in mathematics whose truth or falsehood have not yet been ascertained. I believe that either Fermat's Last Theorem or Goldbach's Conjecture is one of these, and the other has been proved, but at one time neither had been proved. I assume plausible arguments could have been made for and against each. Yet eventually one of them was demonstrated to be true, and eventually the other may be demonstrated to be true or to be false.

Again, you seem to be assuming something that needs to be proved: that all general statements about the world (=all general factual statements) are contingent. Prove it.

If any turn out to be necessary, and we can know these necessary truths, then we can now that they will hold in all cases.

That's not an old chestnut but a philosophical claim.

Assuming that you mean 'induction' in the modern sense and not in the traditional sense which Peikoff and Rand uses (and all philosophers before the late 19th century used), then yes, it is not deductively valid mode of inference, but that's merely because it is not deduction. It can still be inductively valid (though you may, with some logicians, prefer to avoid the terms 'invalid' and 'valid' when talking about induction instead use 'weak' and'strong'). Anyway, that is mainly a matter of terminology. It sounds like you do not agreewith falsificationists and Hume that induction is worthless.

Anyway, one of the few things Hume got right in this area is in seeing that the common belief that deductive connections are necessary and inductive ones absolutely contingent; rather, he saw that necessity is involved in both. So, if there are no necessary factual truths, as he thought, then we cannot do either induction or deduction, in factual matters. Fortunately, he was wrong.

Are you saying that writing a truth table of the truth-functional statement will, if correctly done, prove the statement? I never denied that.

Perhaps you are be assuming that when I claim that such statements are factual that I am denying their necessary falsehood, because you assume that what is necessary (necessarily true or necessarily false) cannot be factual, and you assume that I share that assumption. I do not share it and challenge you to prove it.

The claim that necessary truths are not factual is the most important of all the claims we have been discussing that needs to be proved.

I never denied that, either.

Again, you seem to be assuming that when I claim that such statements are factual that I am denying their necessary falsehood because you assume that what is necessary (necessarily true or necessarily false) cannot be factual, and you assume that I share that assumption, which I do not.

Logical contradictions can be particular, but, more importantly, statements of fact can be either particular or general.

I think you main point is that logical contradictions are not statements about fact. I think you are assuming that logical contradictions are necessarily false and that necessarily false statements cannot be about facts, just as you probably assume that tautologies are necessarily true and that necessary truths are not factual.

But the claim that necessary truths are not factual needs to be proved--and it is the most important of all the claims we have been discussing that needs to be proved.

I never denied that either. Again you seem to be assuming that in claiming that they are factual that I am denying their necessary falsehood because you assume that what is necessary (necessarily true or necessarily false) cannot be factual, and you assume that I share that assumption, which I don't.

If 'so happen to be true' means 'merely so happen to be true', then that is saying that they are always contingent, never necessary. But that needs to be argued for.

I deny this. I say, for example, that "All water is H20" (that is, all water has the atomic structure expressed by that formula) is true and we know it to be true, because we know that being H20 is a necessary fact about water, not just a contingent one. So, to use the Twin Earth example from Putnam I discussed earlier, Twater is not water and cannot be water, because it is not H20.

Similarly, "All gold has the atomic number 7" is true and we know it to be true, because we know that being having that atomic number is a necessary fact about water, not just a contingent one. Now you may think otherwise, because you probably assume that the only attributes of gold that are necessary are the ones contained in the Nominal Definition---that is, the definition that we learn when we first learn the meaning of the term 'gold', such as "a shiny yellow metal" or something like that. However, by such a definition Fool's Gold would have to be counted as a kind of gold. But we never do: we always say that it is just something that is similar to gold.

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## Gregory Browne

If you want to define 'synthetic truth' this way, and define 'analytic truth' as definitional truth, I will accept that, for the sake of the argument, but then you must be accept the consequences of doing so: now you can no longer say that the following are true by definition:

Truths whose denial is a self-contradiction = analytic truths

Truths whose predicate is contained in their subject = analytic truths

because those presuppose

Truths whose denial is a self-contradiction = definitional truths

Truths whose predicate is contained in their subject = definitional truths

which needs to be argued for

(note: I do not deny that all of those the right of the = sign are included among those to the left of it, but I deny that all of those to the left of it are contained in those to the right of it).

See above.

There can be uncertainties even about analytic truths, if we do not know how to prove them (for example, if Goldbach's conjecture and Fermat's last theorem are true, then they are analytic, but at least one of these remains unproven--I forget which; and many theorems in trigonometry were at one time not known.).

But I do take your point that they could be know with certainty if we found the proof.

But what you need to argue now is that no synthetic truth could be known in this way.

It is relevant that in some cases the propositions have not been derived yet, because it shows that the fact that they have not yet been derived does not show that they could not be derived, and so the fact that proofs of some admittedly factual propositions have not yet been so derived does not prove that they could not be.

And deriving universal generalizations about the world from definitions does not imply omniscience. It simply requires that they be necessary truths. And you will admit that necessary truths can be derived from definitions. What you will deny is that they are factual (about the world), because you think that necessary truths can't be about the real world. But you need to prove that.

As I said above, you can define 'analytic truths' as definitional truths, and so you can prove the first part, but now you cannot define 'analytic truths' as 'truths whose denial yields a contradiction', and so you cannot assume that definitional truths are the only truths whose denial yields a contradiction, but rather need to argue for it.

I don't know what your first sentence means---are you saying that you don't know what I mean by 'necessary synthetic truth' or are simply denying it? A necessary synthetic truth is one which is both necessary and synthetic: that is, it cannot be false (and so is true in every universe) but it is not definitional.

Now why do you think that it is impossible for a non-definitional truth to be necessary?

First, my original request was that you prove that only definitional truths are non-falsifiable, so it is only true statements that we are talking about at this point.

Second, you may say that a non-falsifiable statement may be uncertain, but clearly a falsifiable statement is uncertain.

Thirdly, you say that only definitional truths are certain--but that is part of 3, which what you are to argue for: simply asserting it is not enough. What makes you think that only definitional truths are certain?

This assumes that all truths derivable empirically are falsifiable. That assumes the truth of 3 and 4:

3. that only definitional truths are non-falsifiable (i.e., certain, provable with certainty)

4. that definitional truths are not knowable empirically (i.e. from experience)

The claim that definitional truths don't give any information about the world is precisely the claim that I want to argue for.

Yes, I do see that you said "

newinformation about the world", and that is important. Yes, they don't tell you anything about the world that is not already contained in them, but that doesn't mean that they don't tell you anything about the world. And, yes, they don't tell you anything that you don't already know (unless it logically implied in and therefore deducible from what you know) when you learn the meaning of the word, but that does not mean that what you know is not factual.The phrase "the latter" did not refer to any of those examples; as I remember, we were talking about Shallow and Deep Kinds, and those three were examples of statements about the former, whereas it is statements about the latter about which you may be uncertain.

You have doubts about whether we had to discover those two facts? Surely you didn't mean that, or are you actually counting those as definitional truths?

I think we encounter triangles in experience--that is, we encounter things that have triangle surfaces--and then from several such cases we abstract the concept of a triangle, and then we analyze this concept to derive trigonometrical propositions; so such propositions can be proved by analyzing the concept of a triangle, and that this concept is derived from experience. My point is that experience contains all of the information or data you need to ascertain the truth of trigonometrical propositions. Sure, this data must be processed by your rational faculty, but the data is still found in experience and so I say that we should not deny it the label 'empirical' simply because it had to be processed.

I meant to say above "I don't concede that all empirical statements must be falsifiable", and so that is what I want you to prove.

I will have to look at those links.

I believe that Big Bang theory is probably true, because the great majority of scientists do, but I did hear one of who thought that the evidence for it was inadequate. It isn't irrational to doubt it.

I don't know Peikoff's reasons for doubting it. But if he was studying physics in order to prove it, and he came up with arguments based on physics, and not just philosophy to doubt it, you wouldn't object to that, would you.

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## BaalChatzaf

It just so happens that a clear drinkable liquid necessary for sustaining life on this planet has the chemical makeup H20. On another planet in another galaxy with a completely different history of life evolution, this might not be the case. Who knows, somewhere some life forms might like liquid methane or liquid nitrogen which is clear. We don't know. But it is logically possible. The only living things we know about are water based. This need not be necessary. It could be that it so happens to be the case on this planet.

As to gold, it not only has to be shiny and yellowish but we also require that it be ductile and not oxidizable in air. Fools Gold is brittle and not ductile.

In any case we know of no synthetic general statement (universally quantifiable) that must be true in every possible world. So Kant's claim that such a statement exists is at worst false and at best an unsupported speculation. We know his prime candidate, the axioms of Euclid are not synthetic. They are conventional posits (i.e. just plain assumptions). We know that Euclidean geometry is not the only consistent geometry. At the time Kant wrote this, he did not know this to be the case. Kant also claimed Newtonian Physics was not only synthetic but necessarily true. We now know Newtonian physics is false, but close enough to being right to be useful.

As to mathematical truths, we prove statements of the following form: X follows from axioms A. That is an analytic statement based on the definition of "follows from". When we claim (for example) an statement in number theory is true, we are really claiming it either follows from the axioms of the real number system but limited to integers or it follows from the Peano axioms for integers. Some claims are conjectures. For example, the Goldbach conjecture which is not known to be true (i.e. follows from the axioms for integers or from the axioms for real numbers). The Fermat conjecture (miscalled Fermat's Last Theorem) was shown to be true by Wiles in 1993.

Kant's claim that apodictic synthetic statements exist is bogus. His entire analysis underlying -Critique of Pure Reason- comes crashing to the ground.

Ba'al Chatzaf

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## tjohnson

'H2O' is a simple symbolic representation of water, water is

notsimply H2O. To say "All water is H20" is a gross simplification of the situation and should be re-worded to say "water is usually represented by the chemical formula H2O"From wikipedia;

"Water can be described as a polar liquid that dissociates disproportionately into the hydronium ion (H3O+(aq)) and an associated hydroxide ion (OH-(aq)). Water is in dynamic equilibrium between the liquid, gas and solid states at standard temperature and pressure, and is the only pure substance found naturally on Earth to be so."

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## Gregory Browne

I did not say "Water is 'H20'". I said "Water is H20". I assume that, as a student of semantics, you know the significance of the inverted commas and so know the difference between H20 and 'H20', and the difference between water and 'water' (the use-mention distinction): 'water' is a term which refers to water, and 'H20' is a term which refers to H20.

In any case, to avoid any such confusion, I explicitly said what I meant by "All water is H20": namely, I said:

(that is, all water has the atomic structure expressed by that formula).

Now if it turns out that there the formula 'H20' can be used to refer to another chemical stuff as well as to water then you have a real criticism---though only if my example, and not of my point.

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## Gregory Browne

You seem to be missing my point. Perhaps it is because you are unfamiliar with Putnam's Twin Earth example. I discussed it in this forum before but it was some time ago, so for you and anyone else who forgot it or missed it I will restate the main points:

Putnam asks us to imagine that we discover a planet very much like Earth, called "Twin Earth" On there is much that is like Earth (even people speaking a language similar to English, but I will leave out that part as I think I make my points without it). There is a material there which seems to be water. It is a liquid at normal temperatures and pressures, it is tasteless, colorless and odorless, it quenches thirst and it is found in their lakes and rivers (this list may not be exactly Putnam's, but it is close enough for our purposes): in short, it has all the superficial attributes of water). And we start calling it "water". Later, however, we do chemical tests on it and find out it that it is not H20. Putnam then poses the question: is it water, or not. That is, is it a new subkind of water, or just something that it is a lot like water?

The mainstream view, of the Logical Postivists and Ordinary Language philosophers, going back to Hume and still further back to Locke (and Descartes), too, commits them to the view that it is water. This is sometimes called the "description theory of meaning", which says that the meaning of the word is a certain description which we learn when we learn how to use the word, and so anything that fits that description is a referent of the term: in the case of water, whatever has the superficial attributes listed above would be water. As Locke would put it, whatever has the "nominal essence" of water (by which he meant the nominal definition) would be water.

However, Putnam denies that it is water. He says that it cannot be water because it does not have all of the essential attributes of water (by which he means the necessary attributes of water), because of the essence of what he calls "natural kinds" (which I call "Deep Kinds") has more to it than is contained in the description he gave. He says that because he says that the meaning of natural kind terms have an "indexical" (pointing) component: so, for example, it is part of the meaning of 'water' that it refer to stuff like this stuff (said while pointing to a "paradigm"--example--of water)--stuff having the attributes of this stuff, including attributes that may currently be unknown. (At this point is position is similar to Peikoff's: that a concept can include unknown attributes)>

Thank you for giving me an example to prove my case. You say that gold "has to be" not oxidizable by air and ductile, as well as shiny and yellowish. So you are conceding that those two attributes are necessary attributes, and yet those two attributes (or at least the first) are not part of a nominal definition of gold, but rather had to be discovered. So "All gold is not oxidizable by air" is an empirical and factual truth but also a necessary truth.

Now you may wish to say that you did not mean that to say of either "Gold is not oxidizable by air" or "Gold is ductile" that it is a necessary truth. However, if you deny that the latter is necessary, then you have no reason to say that Fool's Gold is not gold. How do you know that it is not gold?

I agree, if we use 'synthetic' as Kant did, to mean a truth whose predicate is not contained in its subject, or a truth whose denial is not a contradictory, because, as I have argued in earlier posts, there are no synthetic truths.

However, if you use 'synthetic' the way Cal does, to mean 'non-definitional' (where the definition is question is a nominal definition), then we do have an example of synthetic necessary truth, Ba'al": see the example above that Ba'al gave me.

Right, under either definition: math is analytic.

As to assumptions being conventional, you are using 'conventional' in an odd sense. In any case, what are these assumptions supposed to be based on?

If we take 'conventional' in the usual sense then, as I have been saying, it is a mistake to think that definitional truths are merely conventional (it is because most of them are about Shallow Kinds that we think so, because Shallow Kinds have concepts so simple that we can learn them when we first learn the meaning of the word, and this misleads us into thinking that they conventional). However, if you disagree, then let's see your argument.

I think in fact Newton's Three Axioms of motion are true and indeed necessarily true, because they follow from the concepts of force and body, just as trignometry follows from the concept of a triangle.

Yes, I know it is commonly thought that Relativity Theory refuted them, and indeed it seeems to have refuted much of the assumptions of Newtonian physics (though I came upon a post on the SOLO site, whose author seems to have been a scientist, arguing that this was a myth), but I don't think it refuted any of the 3 axioms.

What aspects of each of the 3 axioms do you think were refuted?

Note that one of the weirdier aspects of Relativity Theory, the claim that mass increases with acceleration, actually came about because of a refusal to abandon the 2nd axiom. (Although I think that this was not handled right)

One more related point: the Law of Addition of Velocities was not refuted, either.

To truly prove it the axiom itself must be proved, or else be self-evident.

None of which interferes with anything I said, and these examples in fact support one of my points, that to say that a truth is analytic or necessary is not say that it is certain: Fermat's conjecture is a mathematical truth, and so is necessary and analytic, but nonetheless was not certain or even known to be probable until 1993; the same is currently true of Goldbach's conjecture if is true.

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## tjohnson

Does anyone understand this??

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## BaalChatzaf

Wiles proved Fermat's Last Theorem (so-called; it was really a conjecture) in 1993. Between the time Fermat stated it and the time it was proved, it was never proven conclusively but it was generally believed to be true. The attempts at proving Fermat's Last Theorem (so -called; abbr. FLT) turned out to be one of the most fruitful episodes in mathematics. Lots of new mathematics was developed just to prove FLT, for example, the theory of (algebraic) ideals invented by Kummer.

Just to make sure you know what is what FLT states there exists no integer larger than 2 for which the equation

x^2 + y^2 = 0 has non-trivial integer solutions.

Ba'al Chatzaf

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## tjohnson

Thanks Ba'al, but I was being sarcastic (sorry). I thought the statement was pretty well meaningless and I wanted to see if anyone else did as well.

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## Dragonfly

AuthorYou mean: there exists no integer n larger than 2 for which the equation

x^n + y^n = z^n has non-trivial integer solutions.

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## Dragonfly

AuthorAt the moment I have no time for a long discussion, but I'd like to answer two points:

I have serious doubts about the statement that gold has atomic number 7, gold is rather heavy, you know...

Wrong. You shouldn't confuse a mathematical axiom with a Randian axiom, these are

completelydifferent animals. You cannot prove a mathematical axiom, that is exactly why it is an axiom! Neither is there anything self-evident in a mathematical axiom. The only thing you can say is that some axioms will generate more interesting theories than other ones, and for a meaningful theory a set of axioms shouldn't be self-contradictory, but that's all.## Link to comment

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## BaalChatzaf

thanks. I had a brain fart.

Ba'al Chatzaf

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## Gregory Browne

This is an example of a truth that is necessary and analytic, but was once not certain, which shows that necessity is not the same as certainty and analyticity is not the same as certainty.

What exactly do you fail to understand about that?

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## Gregory Browne

Yes. The atomic of gold is 79, rather than 7.

Which of course doesn't affect my point.

Your position is the Aristotelian position on self-evident truths, that they cannot be proved (though Aristotle believed they don't need to be) but this is one of the few cases in which modern logic actually is superior to Aristotelian: modern logic acknowledges that, trivially, every proposition follows from itself, and so can be deduced from itself, and this is true even of a self-evident propositions. So "All bachelors are unmarried" can be proven by being deduced from the self-evident truth "All bachelors are unmarried". Of course this is trivial and boring, and also superfluous, because self-evident truths are self-evident: once you understand their meaning, you see that they are true. So '2 + 1 = 3' can be known be true once one knows the meanings of '2', '1', '3' and of course '+' and "=" and understands the grammar of the sentence.

If mathematical axioms were neither provable nor self-evident they could not be used to prove anything true. You could only prove that a given mathematical proposition follows from the mathematical axiom, which may be what you mean when you say that you are proving the proposition. But if the mathematical axiom is not true and shown to be true then the alleged proof has given us no reason to think that the mathermatical proposition deduced from it is true.

So maybe you are saying that mathematical axioms are not true or false, or are not known to be. Is that your position?

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## Gregory Browne

Some time ago Daniel asked me to present my own summary of Peikoff's argument in the ASD. I was going to do so, but got busy and put it off.

However, then I got to thinking about it: this is a forum on the subject of Peikoff's ASD, and so people discussing it, or at least those who want to attack it, should have read it themselves.

Yet later I read someone say that Peikoff simply asserted his claims. So I decided to resume the project, or at least part of it.

Below is my paraphrase of what seem to be his main points. Later I may recast this in formal argument form, as I did several arguments of myself and others on this forum in earlier posts. But I decided to post what I have now. I have left out most of his summary of the views that he is attacking (though I may include this in a later post), since some have confused the views he is attacking (e.g., that a concept means only its definition) with his own views, and I have left out most of the other historical material.

Page references are to the ITOE, Expanded 2nd ed., 1990

Summary paraphrase of Peikoff's "The Analytic-Synthetic Dichotomy"

“Analytic” and “Synthetic” Truths

Analytic truths are those which can be shown to be true merely by analysis of the meaning of their constituent concepts. 94

A concept means the existents which it subsumes (the units which it integrattes), including all of their attributes 94(98)

Concepts mean existents, including their attributres, not arbitrarily selected subsets of these attribures 98

There is no basis for excluding some of the existents’ attributes from the concept’s meaning 98

Every truth about an existent X reduces, in its basic pattern, to:

“X [which means X, including all its attributes] is one or more of the things which X is.” 100

So when making a statement about X, our choices are to say:

“X is what it is” or

“X is not what it is”

So the choice is between tautology (in a sense) and self-contradiction. 100-1

So, in a sense, no truths are analytic, because no truth can be shown to be true merely by conceptual analysis, without prior discovery of the content of the concept (i.e., the attributes of the existents which the concept refers to);

in a another, sense, all truths are analytic, because the denial of a proposition ascribing an attribute to the existent contradicts the meaning of the concept (since the concept means the existent, including all of its attributes) . 101

‘I married a wonderful man’ does not mean ‘I married a wonderful combination of animality and rationality’. 104

Philosophers separate a thing from its characteristics, but, sensing a connection, say that the thing is one part of the meaning: that meanings have two components: extension and intension. 104

But this distinction is bad, because it separates an existent from its attributes. 105

Necessity and Contingency 106

The Law of Identity entails the Law of Causality, and so a thing cannot act contrary to its identity. Metaphysically, all facts are part of the identity of a thing, and so all metaphysical facts are necessary. Only facts resulting from human free-will choices are not necessary. Metaphysically, the Law of Identity determines how a thing will act, and it has no alternative to doing so. 109

Logic and Experience 112

Knowledge is not acquired by logic apart from experience or experience apart from logic, but by the application of logic to experience. 152

Three versions of this distinction are given below.

1. Logical v. Factual Truths

A dichotomy between truths of logic and truths of experience is analogous to dividing arithmetical truths into summational truths (truths which state the actual sum of a column of figures) and additive truths (truths arrived at by use of the laws of addition). 113-4

2. The Logically possible v. the Empirically Possible

The dichotomy between logical possibility and empirical possibility should be rejected because it requires assuming that a violation of the laws of nature would not involve a contradiction. But that cannot be, because a violation of the laws of nature would require an entity to act in contradiction to its own identity. 115

This dichotomy also presupposes the error of believing that a concept means only its definition.

This dichotomy is often argued for by saying that the empircally possible is conceivable while the logically possible is inconceivable 114-5

But ability to conceive otherwise does not alter the facts. 116

Furthermore, in the serious, epistemological sense of the word, you cannot conceive the opposite of that which you know to be true (apart from statements about man-made facts). If a proposition asserting a metaphysical fact has been demonstrated to be true then it has been shown that the fact is part of the identity of its subject, and so any alternative would require the existence of a contradiction. If a person knows this fact, then only by evading the fact could the person conceive the alternative. 116-117

3. The A Priori v. the A Posteriori

The view that some propositions are a priori---validated independently of experience, simply by analysis of defintions---is mistaken because definitions are based on experience. 117

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## BaalChatzaf

The statement "the sum of angles in a triangle in a Euclidean plan is 180 degrees" is true a priori. No one has ever experienced a triangle. There are no triangles in the physical world. Neither are there points, planes or lines. These things are purely abstract and live in human brains as concepts.

Ba'al Chatzaf

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## Dragonfly

AuthorA mathematical axiom is

by definitiontrue, therefore it is nonsense to talk about "proving" an axiom. A physical model may use some mathematical theory, but the question whether that specific theory (for example a geometric model) can beappliedto a physical phenomenon is independent of the validity of that theory. You shouldn't confuse physics with mathematics. Roughly said, the analytic-synthetic dichotomy corresponds to the mathematics-physics dichotomy.## Link to comment

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## Dragonfly

AuthorExactly. This is a crucial point that has to be understood.

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## BaalChatzaf

I would have preferred to say a mathematical postulate is -assumed- to be true, for the sake of deriving its logical consequences. A mathematical postulate is unlike an -axiom- in the sense that Rand proposed. For example one can assume that more than one line can connect a pair of points. Obviously this is not the Euclidean line that we know and love. This is not the ideal of an infinitely thing thread stretched tight. By the way, straight line is derived from the Anglo-Saxon "strecht linen" which means a stretched linen thread.

That is why we can have Euclidean geometry and an infinite number of consistent non-Euclidean geometries. In point of fact if Euclidean Geometry is consistent then so is Riemannian Elliptical Geometry and Hyperbolic Geometry. That are either all consistent or all inconsistent (within themselves).

Whereas if one denies the axiom -- something exists--, one contradicts the denial since something has to exist to make the denial. That is the difference between an mathematical axiom (or postulate) and what Rand meant by an axiom.

Ba'al Chatzaf

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## tjohnson

We use undefined terms in every language, it is inescapable. In mathematics AND natural language 'point' would be considered undefined, and so one just 'knows' what it means. It is when we start defining words with other words that the difference between mathematics and natural language becomes evident. So if a circle is defined as the locus of points equidistant fro a point called the center we can see at one there is no such thing in nature.

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## Gregory Browne

Ba'al,

Yes. I agree with almost everything you say here. What Cal is talking about are postulates, not axioms (in Rand's sense or in any mathematical sense that I have ever heard of).

I like your point about the stretched linen thread. This should serve to remind us of the empirical origin of even our mathematical concepts.

Similarly, geometry began as more or less what we would call "surveying" today: 'geometry' literally mean 'measuring the earth'.

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## Gregory Browne

You cannot know that a statement is true unless it is either self-evident or provable by derivation (deductively or inductively) from the self-evident. Therefore you cannot know that an axiom is true unless it is self-evident or provable. Therefore you cannot know that it is an axiom unless it is self-evident or provable (note that I am not denying that it will still be axiom: statements can be true without us knowning them to be true).

I think you have axioms confused with postulates (see my last reply to Ba'al). You may assume a postulate p to be true in order see its consequences, or even to prove it false. Then if you can validly deduce q from p you have proved the truth of the conditional statement 'If p then q'. But you will not have proven q, if you have not proven p.

If q is false, and is either self-evidently false or provable as false by deduction from a self-evidently false statement, then you can prove p false (this is a "Reduction ad Absurdum" of p), though proving q true will not prove p true.

Ttruths of physics and truths of mathematics differ in their subject matter, but not in their truth.

I know that the analytic-synthetic dichotomy is supposed to correspond to the mathematic-physics dichotomy, but I deny this--and do so even on your definition of 'analytic truth'--i.e., definitional truth--because I say that Newton's axioms can be derived from definitions of the terms 'body' and 'force', and so, even by your definition, are still analytic.

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## Gregory Browne

As I said in early post, abstraction is a mental process by which selectively focus on certain attributes of something (and so ignore the rest), and the word 'abstraction' can also be used to refer to the result of that process, which is itself a mental entity, a thought or idea. It is common mistake to reify this and so suppose the existence of abstract entities outside of the mind, as Plato did.

So insofar as their it is true that these beings do not exist in the physical world we should say that they don't exist at all, but we have thoughts of them--just as we should say, when we trying to precies that vampires and Sherlock Holmes do not exist at all, but we have thoughts of them (as opposed to saying "Vampires and Sherlock Holmes exist in the mind", which is only true metaphorically: they don't literally exist even in the mind, but our thoughts of them do exist there).

Much more importantly, the fact that there are no perfect triangles in the world does not make that statement true a priori, because our ideas of triangles were derived from experience, even though indirectly: that is, we encountered approximate triangles in experience and acquired the concept of approximate triangles and then idealized the concept to form the concept of perfect triangles. But the material for this concept came from experience, and so it is an empirical concept. (Similarly with your example a perfectly straight line.) And the above statement was derived from this empirical concept. There fore it should be considered to be known by empirical (a posteriori) rather than a priori means.

(Actually, in the case of triangles and other geometrical beings the concept is even more closely tied to reality: even though there are no perfectly triangular surfaces, there are surfaces that

lookperfectly triangular to the unaided eye, and from these we formed the concept of perfect triangles, without even needing to idealize, and then only later, when we measured carefully, did we find that the triangles were not composed of perfectly straight lines.)Here is a simple refutation of your claim: if the fact that there are no perfect triangles in the physical world proved that truths about triangles are not known empirically, then the fact that there are in the physical world no bodies that are not under the influence of any force would prove that Newton's First Axiom is not known empirically, but the latter claim is false, and therefore so is the former.

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## Gregory Browne

Not if we take 'definition' is the broadest sense of the term, to apply to anything that indicates the meaning of a term. So a definition does not have to be verbal: there are ostensive definitions, which involve pointing.

How does know what they mean, without a definition?

Both could be defined, and in math terms normally are defined, and often need to be, because mathematicians often define terms differently from non-mathematicians (which is a very unfortunate habit they share with most other people: when people come up with a new concept they should come up with a new term for it--either a new word or a new combination of old words--but people are frequently lazy about that and grab a term that already has a meaning).

Most words in natural language a defined, and defined verbally.

See my last reply to Ba'al, concering the apriori.

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