The Analytic-Synthetic Dichotomy

[Daniel Barnes]

I wrote:

>Mike, what problems do you think this vague comment solves, and how does it solve them?

Mike replied:

>The same problems you did here below and in the same manner. Meaning not much for either.

I'm serious. What problems does saying mathematics is "grounded" in reality actually solve? And how does it solve them?

[Michael Stuart Kelly]

Daniel,

This is nothing more than epistemological validation. This allows you to know that you can connect mathematical units to chunks of reality and the elements of reality will be able to be handled in that manner (measured, rearranged, charted, etc.).

If mathematics were completely cut off from reality, like a delusion of demons chasing you around, for instance, or completely random numbers and letters, or a made-up language (like idioglossia) that only one person on earth understands, it would not be able to be used for science. It would have no correspondence to reality.

It allows you to say: "There is a mind/reality connection and knowledge is based on reality."

I'm not sure that is a problem. I think it is more of a starting point for knowledge. It is a manner of identifying your cognitive tools.

But even though it was originally tongue in cheek, I would also like a serious answer. To paraphrase you, what problems does saying mathematics is "not grounded" (i.e., produces "unmatched abstractions" leading to "Platonic suspicions") in reality actually solve? And how does it solve them?

For the record, I do think that "unmatched abstractions," if internally consistent, can exist at the present, but as you hinted, physical applications are discovered later. Just because something is one of those "unmatched abstractions" within the context of today's knowledge does not mean that no connection to reality exists. It merely means that it has not been discovered yet.

And further, for the record, just as it is possible to use words to create stories of entities and places that do not exist, I believe it is possible to create fictitious mathematical "entities" that could not exist, but are based on numbers.

Michael

[Daniel Barnes]

Wolf:

>Add ITOE for the connection between sense data and rigorous, noncontradictory definitions.

The ITOE demonstrates no such connection. It is merely vaguely asserted, and there are miscellaneous ad hominems and some handwaving. Nothing more. Her opening blunder, where she violates the Law of Identity with her own theory, gives a good indication of what is to come.

>Presto! A complete system of knowledge based on the syllogism.

Hence, not.

>It is implicit in every post. No one chats in set theory or modal logic.

Yes, everyone chats in syllogisms instead! Apparently they do this 'implicitly' even when they know nothing about syllogisms, or even when their unwitting syllogising is completely incompetent. No doubt one could make the exact same claim about people "implicity" chatting in set theory or modal logic if one so desired.

[Daniel Barnes]

Mike:

>This is nothing more than epistemological validation. This allows you to know that you can connect mathematical units to chunks of reality and the elements of reality will be able to be handled in that manner (measured, rearranged, charted, etc.)

This doesn't answer my question. I will ask it more clearly. What problems prompted thinkers to examine, and then question, the relationship between mathematics and physical reality -and also human consciousness - in the first place? And how do you think merely saying mathematics is "grounded in reality" solves these varied and interesting problems?

[Daniel Barnes]

Ba'al:

> People who are paid to do Logic and who publish in refereed mathematics journals define Logic as the science or discipline of valid inference.

Yes, but perhaps accepting this definition would be problematic for Objectivism, given their claim that induction is valid.

[Michael Stuart Kelly]

This doesn't answer my question. I will ask it more clearly. What problems prompted thinkers to examine, and then question, the relationship between mathematics and physical reality -and also human consciousness - in the first place? And how do you think merely saying mathematics is "grounded in reality" solves these varied and interesting problems?

Daniel,

Survival for starters.

Michael

[BaalChatzaf]

Ba'al:

> People who are paid to do Logic and who publish in refereed mathematics journals define Logic as the science or discipline of valid inference.

Yes, but perhaps accepting this definition would be problematic for Objectivism, given their claim that induction is valid.

Which we know is not the case. Induction does not always lead from true premises to true conclusions. Sometimes induction works, sometimes it doesn't. Imagine the surprise of the bird lovers when they discovered a non-black crow in Australia.

[Wolf DeVoon]

It is implicit in every post. No one chats in set theory or modal logic.

Yes, everyone chats in syllogisms instead! Apparently they do this 'implicitly' even when they know nothing about syllogisms, or even when their unwitting syllogising is completely incompetent. No doubt one could make the exact same claim about people "implicity" chatting in set theory or modal logic if one so desired.

When you use terms like "exact," "people," and "theory," you are using predicate logic in defining those concepts. Surely you know what you are saying and rely on the rest of us to understand as well. The basic structure of a syllogism proceeds from affirmative, negative, universal, and particular propositions. Your use of natural language English is made possible by predicate logic.

W.

[Michael Stuart Kelly]

Induction does not always lead from true premises to true conclusions. Sometimes induction works, sometimes it doesn't. Imagine the surprise of the bird lovers when they discovered a non-black crow in Australia.

Bob,

Imagine an even greater surprise on discovering a non-black crow that weighs two tons, has hair instead of feathers, has two heads, seven feet and five wings. Now that would be something!

Induction would be completely shot all to hell.

Michael

[BaalChatzaf]

Induction does not always lead from true premises to true conclusions. Sometimes induction works, sometimes it doesn't. Imagine the surprise of the bird lovers when they discovered a non-black crow in Australia.

Bob,

Imagine an even greater surprise on discovering a non-black crow that weighs two tons, has hair instead of feathers, has two heads, seven feet and five wings. Now that would be something!

Induction would be completely shot all to hell.

Michael

A crow (whatever its color) is a bird. A critter with five wings is not a bird. Birds have the same bauplan as we do. Two upper limbs two lower limbs and one head.

Anyway you want to analyze it, induction does not always produce a true conclusion (a universally quantified generality) from a limited set of correct instances. The only way to guarantee an induction is correct is to exhaust the domain of individuals over which the induction is done. In general, this is not possible.

Ba'al Chatzaf

[Daniel Barnes]

Mike:

>Survival for starters.

I hardly think it was problems of "survival" that prompted the thinkers of history to ask questions about mathematical epistemology!

If you don't grasp at least some of the problems, it won't make sense debating anyone's proposed solutions.

[Michael Stuart Kelly]

A crow (whatever its color) is a bird. A critter with five wings is not a bird. Birds have the same bauplan as we do. Two upper limbs two lower limbs and one head.

Bob,

Uh oh. How do we know that about birds? Deduction? Induction?

(You already know I am going to say induction.) Seems to me there is a kind/degree issue here.

Incidentally, the whole argument about white crows or black swans does not invalidate induction. It invalidates logic in general for that issue simply because that issue was flawed as stated to begin with. It was not built on fundamentals. "Black" is a defining characteristic of "crow," since most all crows are black (ditto for swans and white), but it does not eliminate the possibility of other colors, just as "midget" does not fall outside of "human being." Part of the very essence of living things (borne out by observation and induction) is that all members of all species are individual and vary in some degree (ranging from enormously to minor) from the other members of the same species.

Since the meaning of life itself includes this variation of individuals within species, the whole attempt to prove somehow that crows (or swans) are different from all other life forms at the root through induction shows the silliness of the exercise, not the failure of induction. The premise is flawed. A simple syllogism blows the whole thing out of the water at the get-go (through deduction, I might add):

Premise A: All individual members of all species vary from one another to different degrees.

Premise B: All individual crows are invariably black.

Conclusion: Crows are not members of any species.

Boom.

We have a problem and it ain't induction.

"Bird" is the genus, so that will not change for crows. What can change is the differentia as one learns more, whether by observation or through the exercise of some form of logic. This is standard Objectivism—knowledge is hierarchical and contextual, thus it is subject to change as the hierarchy or context changes. I presume you know that.

Mike:

>Survival for starters.

I hardly think it was problems of "survival" that prompted the thinkers of history to ask questions about mathematical epistemology!

If you don't grasp at least some of the problems, it won't make sense debating anyone's proposed solutions.

Daniel,

Of course it was. They didn't use the word "epistemology" at the start of recorded history, but they sure as hell applied their math.

As I understand it, you wish me to provide a mathematical problem showing "the relationship between mathematics and physical reality -and also human consciousness - in the first place." In the first place? You want me to use a math problem to prove the validity of math to reality during the initial use of it? Don't you already have to be using math in order to do that? That is circular as all get-out.

To properly answer this question, the problem had to come from reality, not from math. (Applied math is still math.) You must prove what is solves in reality. Survival is one hell of a problem for 100% of mankind in reality, especially since 100% end up dying, but they can postpone the arrival of that moment with knowledge, especially when math is included.

If you have another question, that's another issue. I answered your question as I understood it. Besides, I stated clearly that I did not think of this as a problem, but as a validation for knowledge: a starting point. See below. I also asked a question of my own that has not been addressed so far.

I'm not sure that is a problem. I think it is more of a starting point for knowledge. It is a manner of identifying your cognitive tools.

But even though it was originally tongue in cheek, I would also like a serious answer. To paraphrase you, what problems does saying mathematics is "not grounded" (i.e., produces "unmatched abstractions" leading to "Platonic suspicions") in reality actually solve? And how does it solve them?

Michael

[Gregory Browne]

You still haven't proved that non-definitional truths are synthetic, because you need to prove that it is only definitional truths whose denial yields a contradiction.

A proposition that is not analytic cannot yield a contradiction upon denial, as it cannot be 100% certain, in contrast to an analytical proposition.

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?? 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.

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.)

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.

Yes, you do have to argue for what you claim, unless it is self-evident or your opponent concedes it, because you are claiming the existence of these dichotomies: it is not up to your opponents to prove that the dichotomies do not exist

It's just the other way around. Peikoff argues that the dichotomy doesn't exist; I've shown the flaws in his argument, that's all, I don't have to prove anything.

If you had shown them, you would have proved a lot, but you have not, because your premises are neither self-evident nor conceded by person you are debated nor conceded the author of the article you are critiquing. Your usual pattern is defend one of the dichotomies Peikoff attacks, such as the analytic-synthetic dichotomy, by making an argument the presupposes the validity of one of the other dichotomies he attacks or some other claim he rejects.

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.

Now lest you think that Peikoff and I agree with you on nothing and would concede nothing, I will list some claims that we agree with you on:

6. that definitional truths are analytic

7. that definitional truths are necessary (express necessary facts, as Peikoff would put it)

8. that definitional truths are certain and non-falsifiable

9. that non-definitional truths are knowable empirically

10. that non-definitional truths are know factually

11. that necessary, analytic truths are not all trivial

It can be self-contradictory even if you cannot be sure of its truth.

But in that case you don't know that it is self-contradictory, so you can't say that the statement is self-contradictory, in contrast to self-contradictory analytical statements, where you can categorically say that they are self-contradictory, they don't need any empirical evidence for that. That is the essential difference that Peikoff and you can't argue away.

I don’t deny that there is a difference: I just deny that it is the difference you think it is.

The difference is between truths about Shallow Kinds, on the one hand, and truths about Deep Kinds, Narrow Classes and individuals, on the other. Regarding the former, you will have certainty if you analyze the concepts right to come up with your basic truths, and then make deductions without making any mistakes. 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, 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, and you’d have to discover that all ice floats on water—if it did, which in fact it does not, which also had to be discovered.

A statement can be logically true, while it doesn't refer to a fact in reality. If I define "oil" as the solid form of water, the statement "oil is a solid" is an analytical truth. Of course this doesn't at all correspond with what we know of what we commonly call "oil" and "water", therefore such a definition would be useless as it contradicts common usage. It is much more efficient to use definitions of physical things that reflect our empirical knowledge about those things, we prefer that the analytical truth doesn't contradict our empirical knowledge. But that doesn't alter the fact that such a statement is true independently of our empirical knowledge. If we acquire new knowledge that no longer corresponds to that analytical statement, we'll have to adapt our definition(s), which has happened often enough in science.

First, we should not say that 'Oil is solid' is true, because the definition is simply incorrect as expression as of the meaning of the English word 'oil' (Peikoff and I do believe that there are such things as wrong definitions.

But, that being said, I say that it is minor point. More important is the second point: if we modify our language to let 'oil' refer to solid water, then 'Oil is solid' will refer to a fact in reality: the fact that ice (which we are calling 'oil') is solid.

However the above facts do not require us to conclude that truths about Shallow Kinds are not known by empirical means: we can still say that all concepts, including those of triangles, bachelors and ice, are empirical, and if that is true we can say that truths discovered by analysis of those concepts, such as the truths of trigonometry discovered by analysis of the concept of triangles, are known empirically.

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).

But this matters little to me, because, as I have said, scientists are not authorities in this field. It is you who have been invoking their authority. Let's use our own arguments for our claims.

Of course scientists are authorities in this field, perhaps not in all the linguistic niceties, but certainly in the foundations of physics and mathematics, of which most philosophers don't know anything (Rand and Peikoff included). Only philosophers with a solid scientific background like for example Bernard d'Espagnat can be taken seriously in this regard.

You are making an Argument from Authority: you are saying that we ought to believe in these dichotomies because all or most scientists believe in them, and they are authorities in the matters, and so we should take their word for it.

No, you are making an Argument from Authority, by saying that scientists are not authorities in the field,

One does not make an Argument from Authority merely by denying that someone else is an authority: for example, if one shows that a doctor called by an opposing lawyer as medical witness in court case lacks a medical degree, or has never worked in the specialty in which the doctor is called as a witness, then one is attacking the doctor's claim to be an authority, but one is not oneself making an Argument from Authority thereby.

implying that philosophers are,

Well, if anyone is an authority on logic, logicians (who are a kind of philosopher) would be authorities, but I am not sure even they are, and most would not rest any claims about logic on their credentials as authorities, and I certainly am not resting my arguments any claim to be authority.

For that matter, neither is Peikoff.

so the opinion of scientists is in your view of little importance. I merely deny that.

I certainly don't think it is of little importance. I just say that they are not authorities on this, and so we should not accept that what they say about these topics is true just because they say so, and so you should not refuse to argue for your premises by saying that they are what most or all or almost all scientists believe, or argue for them by appealing to what they believe as premises.

However, an appeal to the authority which asks us to believe what most scientists believe about general, basic scientific principles, including principles of general scientific methodology, and to believe it merely on the ground that most or all scientists believe it, is invalid. The basic principles of science, including those of scientific methodology, are principles that any intelligent person with a modicum of education should be able to grasp and analyze intelligently.

And that is a big error. I'm not saying that every scientist is an authority on scientific methodology, but I do say that a philosopher without a solid background in science is definitely not an authority,

As I say, these are logical topics, and if anyone is an authority on logic (which I am doubtful about) it is logicians, but I'm doubtful that even they are authorities, and I certainly don't rest my claims on credentials as an authority.

and certainly not Peikoff, in view of all the nonsense he has said about science (which I've documented elsewhere on this forum).

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).

Again, 'does not depend on reality' is a little vague, but I will assume it means 'true in all possible worlds, i.e., necessary'. This then is the classic argument of some Logical Positivists such as Wittgenstein, who said that necessary truths are not factual (i.e., not about the world) because they are true no matter how the world is. However, being true no matter how the world is, i.e., being true in all possible worlds, does not mean that it does not say anything about the actual world---on the contrary, it means that it says something about each and every possible world, including the actual world.

No. Its referents may refer to things in the actual world, but not necessarily. With the usual definition of unicorn the statement "a unicorn has a horn on its head" is an analytical statement, or in your words, it's true in each and every possible world. But does it say anything about the actual world? We can calculate the volume of a 254306-dimensional sphere embedded in 254307-dimensional Euclidean space, which also forms an analytic statement. Does it refer to anything in the actual world?

Yes: those statement do say something about the actual world (as well as every other possible world). In the case of the unicorn statement it says that nothing without a horn is a unicorn, and so if anyone attempting to prove the existence of unicorns brought in some hornless animal their claims should be rejected immediately.

No, the statement doesn't say anything about the actual world, even if it uses a referent (the horn) from that actual world, as the unicorn doesn't exist in the actual world.

It doesn't have to refer to an actual thing or any actual event to be about the world: it can say something about what is or is not possible in that world, and so be about the world. Two examples:

1. Suppose I say of some poisonous liquid "If you drink that then you will be very ill", I am not saying that you will drink that or that you will very ill. Yet I am saying something factual, someting about the world, even if you never drink it. I am saying something about a dispositional attribute of the liquid.

2. As I said before, Newton's First Axiom describes how a body will behave when it is not acted on by any forces, but such state never occurs, because some force is always acting on any body. The truth describes what would happen in an idealized state. Yet you would not deny that the Axiom is a factual truth.

An analogous but more complicated statement could be made about the second, assuming that a sphere with more than 3 dimensions is even logically possible.

It is certainly logically possible. And I wonder how you could make an analogous statement, as in this case the proposition doesn't contain any referent from the real world.

The concept is composed of elements from the real world.

I wasn’t talking about applicability in so many words, but if you want me to address it, I will: how could a theory be applicable to the world if it is not “factual” (i.e., “about the world”)?

Because there is an important distinction between the mathematical theory and the physical theory. The physical theory is about the world, it describes phenomena we observe and it can make correct predictions. It makes extensively use of mathematics in its models, but that doesn't make the mathematical theories dependent on reality. The physical theory can be falsified, the mathematical theory not. As I mentioned earlier, Rand decided that the concept "imaginary number" was a valid concept, while it could be applied to physical models. But the possibility of applying a mathematical concept to practical purposes has nothing to do with its validity - applicability is not the same as validity.

The claims you make before you mention Rand are again all part of what you are trying to prove, and so cannot be used in any argument for them.

[Gregory Browne]

It will clarify things in this debate to ask whether you are talking about overall similarities between things or similarities in a respect (i.e., in respect to a given attribute). The latter is what is mainly important, but modern philosophers tend to think only of the former.

If by “overall similarities” you mean a general likeness or resemblance, I’ve been talking about the latter, ie ‘similarity’ in terms of an attribute. But as I understand it, overall similarities between things also have a bearing on this problem, because if one rejects real universals surely the next step is to discover some other reason for why we group some things and not others.

I’m not sure about your second comment. I note your comments about trope nominalism in another post, but I would have thought that conceptualists and other types of nominalists would be interested in ways of grouping things in terms of some type of general resemblance. Am I missing something here?

Brendan

Conceptualists and nominalists don't have to group things on the basis of general resemblance. That seems to presuppose that there is only one right way to group or classify things, but that is not true and conceptualists and nominalists usually deny it. Any individual is a part of very many groups or classes (some would say an infinity), and not just higher and lower classes in one classification, but in multiple classifications: we can have cross-cutting classifications. For example, we could classify the mammals into Linnaean taxa, or divide them into terrestrial, aquatic and flying mammals, or divide them into carnivorous and herbivorous mammals, etc. Different classifications have more practical value for different purposes. There may be one classification of a given domain (i.e., the group of things that we are classifying) that is overall the best for most purposes (for example, many think this of the Linnaean system in biology) but in other domains there may not be.

[Gregory Browne]

So you think that a characteristic of astrology must be a characteristic of astronomy, while the former is the "background" of the latter?

Does a contradiction exist? Does nothing exist?

Astrology wasn't really the background to astronomy. It was an application of early astronomy to predicting events that affect humans.

Alchemy, on the other hand, seems to have just been primitive chemistry.

[Brendan Hutching]

Different classifications have more practical value for different purposes.

I take your point, although I don’t follow your comment about modern philosophers focusing mainly on overall similarities between things. Most general commentaries I have read about universals tend to deal with single, often perceptual, attributes rather than the more general and multiple classifications.

What is the significance of modern philosophers focusing on overall similarities?

Brendan

[Daniel Barnes]

I wrote:

>I hardly think it was problems of "survival" that prompted the thinkers of history to ask questions about mathematical epistemology!

Mike replied:

>Of course it was. They didn't use the word "epistemology" at the start of recorded history, but they sure as hell applied their math.

Well, ok. Who were the thinkers of history that thought the chief problems of mathematical epistemology was "survival?"

Oh, they're pre-historical? Then how do you or I or anyone else know what they thought? Look, instead of this freestyle jive, why don't you just stick to thinkers within the last 2000 or so years, and some of the more well-known problems in the standard intellectual tradition? That is obviously what I meant, not the speculative matho-epistemological discussions that may or may not have been conducted over a few bevvys in the Lasceaux Caves after the monthly Mammoth hunt. Please.

>As I understand it, you wish me to provide a mathematical problem

Of course not. See above.

>To paraphrase you, what problems does saying mathematics is "not grounded" (i.e., produces "unmatched abstractions" leading to "Platonic suspicions") in reality actually solve?

The very problems that you are blissfully unaware of, it seems. Maths is of course applicable to reality in some respects. That is one of the most interesting problems to begin with, right there in front of your nose!

[Michael Stuart Kelly]

Daniel,

We have reached an impasse again.

I still did not understand your question nor your answer. They are very vague.

You do awfully well at complaining, though. I just wish there were a bit more clarity so I could see what you really want.

Michael

[Daniel Barnes]

Mike:

>I still did not understand your question nor your answer. They are very vague.

My question was: "What problems prompted thinkers to examine, and then question, the relationship between mathematics and physical reality -and also human consciousness - in the first place? And how do you think merely saying mathematics is "grounded in reality" solves these varied and interesting problems?"

I restricted this question to only those thinkers in recorded history...

How much clearer and simpler could I make it?

>You do awfully well at complaining, though.

Is it any wonder?

[Gregory Browne]

Different classifications have more practical value for different purposes.

I take your point, although I don’t follow your comment about modern philosophers focusing mainly on overall similarities between things. Most general commentaries I have read about universals tend to deal with single, often perceptual, attributes rather than the more general and multiple classifications.

What is the significance of modern philosophers focusing on overall similarities?

Brendan

I have had the impression that most 20th century philosophers in the English-speaking world (roughly "analytic philosophers") tended to be nominalists of the kind who said that our use of general terms was grounded in over-all similarities. But it may be that that view is not as widespread as I thought, which would be good to hear.

There are many realists writing in recent decades, and indeed maybe most writers on universals are realists, which is not the same as saying most philosophers are.

A good general survey, from the 90s, is D. M. Armstrong's Universals: An Opinionated Introduction.

[Gregory Browne]

Wolf,

Rand's definition of logic is given on page 1016 of Atlas:

"Logic is the art of non-contradictory identification."

She remarks also on that page:

"Logic rests on the axiom that existence exists."

It follows, I notice, that it is not logically possible that nothing exists. It is not logically possible that existence might not have been. One logician, David Bostock, remarks to the contrary in his Intermediate Logic (Oxford 1997). He maintains that "it is a possibility that nothing should have existed at all" (page 354). The two opposing views, Rand's and Bostock's, commend different ways of developing modal logic. I won't pursue that just now.

Let me indicate, instead, a little more of Rand's conception of logic. Her definition of logic as the art of non-contradictory identification is made in the context of having cast consciousness as identification. In this conception, logic is embedded in wider processes of identification, all of them relying on Rand's thesis that no existents are without identity.

I would be very interesting in hearing what you have to say about what way of developing modal logic Rand would commend and how it would contrast with the way commended by Bostock.

[Gregory Browne]

Google <modal logic> and Google <first order logic>.

Hmm. I would say Google <syllogism 14 valid>

Aquinas' Commentaries are the clearest presentation IMO.

It's beginning to bug me that critics are trotting out set theory to disprove or obliterate predicate logic. If you wander down the modal logic path, you'll end up with Hosper's doctrine of 'necessary and sufficient,' which is played dues wild. Conceptual knowledge is Aristotelean predicate - sometimes called Classical - logic. Add ITOE for the connection between sense data and rigorous, noncontradictory definitions. Presto! A complete system of knowledge based on the syllogism.

It is implicit in every post. No one chats in set theory or modal logic.

W.

I am glad to hear someone praise Aristotelian logic as compared to modern logic: modern logic has done a terrible job regarding conditionals. And this spills over into their account of the predicate logic of universals.

Modal logic, however, has its place, as long as we disentangle the necessary-contingent dichotomy from the others in the ASD, draw it in the right place, and throw out de dicto necessity.

Otherwise, to be consistent we have to remove these terms from our vocabulary: 'necessary', 'possible', 'impossible', 'can', 'cannot', 'could', 'could not', 'must', 'has to', 'have to', and perhaps others.

[Gregory Browne]

Dragonfly,

Don't we have to learn how to count before we learn higher math? And don't we start by counting things we observe, like fingers and toes?

That's one connection to reality at the start.

Michael

Technically, when you use numbers to count things it is called applied math, albeit it's a very simple application.

I am doubtful that counting is applied math. It doesn't seem to apply any of the branches of math, even arithmetic. It is seems rather to be an assigning of numerals (which are symbols--verbal or non-verbal, written, spoken or otherwise--that stand for numbers) to things, as part of a procedure in which the last number counted is what we say is the number of the things we were counting. It is more of foundation for math than an application of it.

Edited June 18, 2007 by Greg Browne

[BaalChatzaf]

I am doubtful that counting is applied math. It doesn't seem to apply any of the branches of math, even arithmetic. It is seems rather to be an assigning of numerals (which are symbols--verbal or non-verbal, written, spoken or otherwise--that stand for numbers) to things, as part of a procedure in which the last number counted is what we say is the number of the things we were counting. It is more of foundation for math than an application of it.

The operation of counting is part of what cardinality is. Set theory had to be invented (by Cantor and others) to produce a complete definition of cardinality and ordinality.

Ba'al Chatzaf

[Michael Stuart Kelly]

My question was: "What problems prompted thinkers to examine, and then question, the relationship between mathematics and physical reality -and also human consciousness - in the first place? And how do you think merely saying mathematics is "grounded in reality" solves these varied and interesting problems?"

I restricted this question to only those thinkers in recorded history...

Daniel,

That is the final form your question took and the restriction only applied at the end. That is not how the question started. It actually started as a sort of sardonic rhetorical question. Then it started getting serious. Here is the development.

Mike:

>That couldn't possibly be because math is initially grounded in reality, could it?

Mike, what problems do you think this vague comment solves, and how does it solve them?

I'm serious. What problems does saying mathematics is "grounded" in reality actually solve? And how does it solve them?

What problems prompted thinkers to examine, and then question, the relationship between mathematics and physical reality -and also human consciousness - in the first place? And how do you think merely saying mathematics is "grounded in reality" solves these varied and interesting problems?

I hardly think it was problems of "survival" that prompted the thinkers of history to ask questions about mathematical epistemology!

Who were the thinkers of history that thought the chief problems of mathematical epistemology was "survival?"

Oh, they're pre-historical? Then how do you or I or anyone else know what they thought? Look, instead of this freestyle jive, why don't you just stick to thinkers within the last 2000 or so years, and some of the more well-known problems in the standard intellectual tradition? That is obviously what I meant...

So this question has developed from one of epistemology to one of history of philosophy. I will not have a decent answer for you about the history of philosophy until I study more of it. As you can see from your opening question and how it slowly developed (especially the final part going from "thinker," which obviously prehistoric man had to be in order to evolve into a more advanced thinker, to "thinkers in recorded history"), I had good reason to believe you were discussing epistemology and not history. I think this was the source of misunderstanding.

Michael