Question for old-timer's: Peikoff's view on certainty

[merjet]

I wonder: Do you (or any of the other mathematically informed members) know of any source in the general literature on issues of mathematics and truth where this suggestion is raised and discussed? Have you done anything further with it yourself?

I see that Stephen Boydstun responded to your first question. I have not developed it further. Besides some propositions in mathematics, being correct but not true applies to fiction. 'Hank Rearden had an affair with Dagny Taggart in the novel Atlas Shrugged' is true. 'Hank Rearden had an affair with Dagny Taggart' is neither true nor false, but correct.

I was also going to point you to Stephen Boydstun's piece about Ruth Milliken and truth theory, but he beat me to it.

[Brant Gaede]

I see that Stephen Boydstun responded to your first question. I have not developed it further. Besides some propositions in mathematics, being correct but not true applies to fiction. 'Hank Rearden had an affair with Dagny Taggart in the novel Atlas Shrugged' is true. 'Hank Rearden had an affair with Dagny Taggart' is neither true nor false, but correct.

Is this paragraph true or correct? Honestly, it seems language is being squeezed to death here.

--Brant

Edited August 26, 2007 by Brant Gaede

[Daniel Barnes]

Merlin:

>Daniel, I realize you are extremely eager to dismiss anything Ayn Rand said as banal verbalism, but you could have noticed the difference between her two similar phrases simply by reading her next sentence: "you have used all of the knowledge available to you and have not indulged in any evasion."

Hi Merlin,

Well, I don't really see how that additional sentence makes much difference. And Rand writes redundantly quite often, especially in critical junctures in her thought. And I don't see the sentences as so much "similar" as equivalent: to "include everything relevant" is the same thing as to "omit nothing relevant", and both take place within the context of your knowledge, without evasion.

Not everything she said was verbalistic, true, but a damn sight more is than is usually recognised. So if I harp on overmuch it is because this general point is not made enough.

For instance: Do you think that particular para would usually have made it past the sub-editors if it was another thinker? If Wittgenstein had said it, would it have gone into a "Wittgenstein Answers" volume? I am not sure it would have. Contrary to recieved opinion that people are unduly hard on Rand, IMHO she gets way too soft a ride. Or, rather, people tend to criticise the wrong things about her.

>As for the banality that Ba'al raises, I suggest caution. It comes mighty close to "no truth w/o omniscience."

The situation from a Popperian perspective is that you can know the truth w/o omniscience, but you just can't know that you know it.

>For anyone interested I address the coherence theory of truth in Volume 1, Number 5 and the Objectivist theory of truth in Volume 1, Number 6 here:

Thanks Merlin, looks most interesting.

Edited August 27, 2007 by Daniel Barnes

[Ellen Stuttle]

Lest Stephen and Merlin think I didn't notice their replies to my inquiries, this is just a quick note to say "Thanks."

I'm eager to look into Ruth Millikan's work, but I'll probably have to wait until next week -- lots scheduled here this week.

I'm sort of bemused at discovering that Millikan is on the faculty at U. Conn. Only about 40 minutes to an hour's drive from here -- and Larry has a lot of contacts with the U. Conn physics faculty, so once-removed introductions would be easy. Funny thing is, I think Ruth Millikan might even be, judging from her photo, a woman I noticed at the U. Conn. physics colloquium featuring Dick Lindzen in March. (Larry had arranged the event and was announcing and I attended -- a wonderful near 12 hours, including lunch, discussion sessions with physics students and faculty, the colloquium itself, then dinner with Lindzen and some of the physics profs and wives/friends; Lindzen has one of the best physics minds it's ever been my joy to encounter.) There was a woman whom I noticed in the audience -- she seemed like a faculty person, not one of the sprinkling of physicists' wives/friends who were attending. The general appearance was much like the photo of Ruth Millikan. What a "small world," I shall think, if it turns out she was the person I'm thinking of.

I like so much the sound (from the glimmers I've gleaned so far) of the work she's doing...

Ellen

___

[Dragonfly]

Is this paragraph true or correct? Honestly, it seems language is being squeezed to death here.

Indeed. After all it's just a matter of definition of the word "true"/"truth", and I don't see why we should use a different word for mathematical statements. The statement "2 + 2 = 4" is not a true statement but a correct statement? Seems rather awkward to me. Of course there is a difference between a mathematical truth and truth as correspondence to reality, which is related to the difference between analytic and synthetic statements, but I have no problem with that distinction...

[Ellen Stuttle]

Is this paragraph true or correct? Honestly, it seems language is being squeezed to death here.

Indeed. After all it's just a matter of definition of the word "true"/"truth", and I don't see why we should use a different word for mathematical statements. The statement "2 + 2 = 4" is not a true statement but a correct statement? Seems rather awkward to me. Of course there is a difference between a mathematical truth and truth as correspondence to reality, which is related to the difference between analytic and synthetic statements, but I have no problem with that distinction...

To me, saying "correct" instead of "true" as regards mathematical statements doesn't seem awkward but instead a convenient way to track the "difference between a mathematical truth and truth as correspondence to reality," instead of the cumbersomeness of having to explain the analytic/synthetic distinction -- and the hassles of the resultant arguments with people who don't get the distinction. ;-) Seems to me, just saying "correct" as regards mathematics would simplify discussion. (There might of course be disadvantages which I haven't thought of.)

Ellen

___

[Dragonfly]

I guess it's a matter of linguistic taste...

[BaalChatzaf]

Is this paragraph true or correct? Honestly, it seems language is being squeezed to death here.

Indeed. After all it's just a matter of definition of the word "true"/"truth", and I don't see why we should use a different word for mathematical statements. The statement "2 + 2 = 4" is not a true statement but a correct statement? Seems rather awkward to me. Of course there is a difference between a mathematical truth and truth as correspondence to reality, which is related to the difference between analytic and synthetic statements, but I have no problem with that distinction...

I suggest the following usage:

true --- in correspondence with fact

correct -- in accordance with a set of rules. The term valid might also apply here.

definitions are simply posits for word or term usage.

Ba'al Chatzaf

[merjet]

Is this paragraph true or correct? Honestly, it seems language is being squeezed to death here.

Indeed. After all it's just a matter of definition of the word "true"/"truth", and I don't see why we should use a different word for mathematical statements. The statement "2 + 2 = 4" is not a true statement but a correct statement? Seems rather awkward to me. Of course there is a difference between a mathematical truth and truth as correspondence to reality, which is related to the difference between analytic and synthetic statements, but I have no problem with that distinction...

I don't know if this is a response to what I posted earlier or Ellen Stuttle's latest. Regardless, I have not said nor do I hold that all mathematical statements should be considered correct but not true. Some, such as 2 + 2 = 4, are true. It is empirically demonstrable, i.e. it corresponds to reality.

By the way, I took Brant Gaede's remark to be about what I wrote about a bit of Atlas Shrugged. Would he have said the same if my example had been 'God created the world in 6 days and rested on the 7th'? This is correct per the Bible, but it is fiction.

Edited September 11, 2007 by Merlin Jetton

[Robert Campbell]

Ellen,

I heartily recommend Ruth Millikan's work to you. She has one of the better takes on evolutionary psychology. If you can visit her at UConn, you should do it.

I saw her in the audience at a session that featured one of her UConn colleagues, at the American Philosophical Association meetings last December. She has a hearing aid these days, and walks with a cane.

Robert Campbell

[Brant Gaede]

By the way, I took Brant Gaede's remark to be about what I wrote about a bit of Atlas Shrugged. Would he have said the same if my example had been 'God created the world in 6 days and rested on the 7th'? This is correct per the Bible, but it is fiction.

It is also true per the Bible. The real question is is the Bible fiction? If the Bible is essentially fiction--it is-- then what is true per the Bible is merely a subcategory of this fiction. Now, if I shoot down someone trying to murder me my action is correct. It's true if it actually happened. It is also true that it is correct. Again, it's hierarchy, IMHO. The idea of a basic equivalence--weight--even with different meanings is an unnecessary burden for reader and writer with respect to these two words.

--Brant

[Ellen Stuttle]

Is this paragraph true or correct? Honestly, it seems language is being squeezed to death here.

Indeed. After all it's just a matter of definition of the word "true"/"truth", and I don't see why we should use a different word for mathematical statements. The statement "2 + 2 = 4" is not a true statement but a correct statement? Seems rather awkward to me. Of course there is a difference between a mathematical truth and truth as correspondence to reality, which is related to the difference between analytic and synthetic statements, but I have no problem with that distinction...

I don't know if this is a response to what I posted earlier or Ellen Stuttle's latest. Regardless, I have not said nor do I hold that all mathematical statements should be considered correct but not true. Some, such as 2 + 2 = 4, are true. It is empirically demonstrable, i.e. it corresponds to reality.

[deleted para.]

Merlin,

It's both correct and true -- from the only example I know of, the article a paragraph of which I quoted (full passage Here)-- that you weren't speaking of all mathematical statements, instead only of "[m]any propositions in higher mathematics." I had started to wonder, however, prior to reading the paragraph quoted, if "correct" might not be a better way to think of 2 + 2 = 4 than "true." What started me down this line of thinking was that 2 + 2 = 5 would be correct if the meaning of "4" and "5" were different than our English meanings. And then I started to wonder if the rule connecting the terms, thought of as terms, is "true" in the sense of "corresponding to reality."

That //// presents // and // items, yes. But what's the status of the symbolic rule connecting linguistically counted items?

I'm only explaining this to correct (different meaning) your possibly thinking I'd thought you were talking about all mathematical statements. I understood that you weren't.

Ellen

___

[Ellen Stuttle]

If you can visit [Ruth Millikan] at UConn, you should do it.

Would you believe...this fall semester she's occupying the Belle Van Zuylen Utrecht University Chair, according to her CV. (Utrecht, I believe, is Dragonfly's alma mater.)

Ellen

___

[BaalChatzaf]

Merlin,

It's both correct and true -- from the only example I know of, the article a paragraph of which I quoted (full passage Here)-- that you weren't speaking of all mathematical statements, instead only of "[m]any propositions in higher mathematics." I had started to wonder, however, prior to reading the paragraph quoted, if "correct" might not be a better way to think of 2 + 2 = 4 than "true." What started me down this line of thinking was that 2 + 2 = 5 would be correct if the meaning of "4" and "5" were different than our English meanings. And then I started to wonder if the rule connecting the terms, thought of as terms, is "true" in the sense of "corresponding to reality."

That //// presents // and // items, yes. But what's the status of the symbolic rule connecting linguistically counted items?

I'm only explaining this to correct (different meaning) your possibly thinking I'd thought you were talking about all mathematical statements. I understood that you weren't.

Ellen

___

Ellen, have you studied Modal Logics?

Ba'al Chatzaf

[Ellen Stuttle]

Ellen, have you studied Modal Logics?

Studied, no; heard chit-chat about, yes. I don't know enough to understand why you asked that in the particular context.

Ellen

___

[BaalChatzaf]

Ellen, have you studied Modal Logics?

Studied, no; heard chit-chat about, yes. I don't know enough to understand why you asked that in the particular context.

Ellen

___

Modal logics broaden the application of logic beyond just true and false. They get into ideas such as possibility, necessity, knowability and relevance. There are even logics (paraconsistent logics) that limit the damage caused by contradictions.

You seem to be very well attuned to logical issues and you might find it useful to study some extensions of logic -way- beyond the usual classical domains.

Ba'al Chatzaf

[tjohnson]

I don't know if this is a response to what I posted earlier or Ellen Stuttle's latest. Regardless, I have not said nor do I hold that all mathematical statements should be considered correct but not true. Some, such as 2 + 2 = 4, are true. It is empirically demonstrable, i.e. it corresponds to reality.

I would be very interested to see you empirically demonstrate that 2+2=4.

[BaalChatzaf]

I don't know if this is a response to what I posted earlier or Ellen Stuttle's latest. Regardless, I have not said nor do I hold that all mathematical statements should be considered correct but not true. Some, such as 2 + 2 = 4, are true. It is empirically demonstrable, i.e. it corresponds to reality.

I would be very interested to see you empirically demonstrate that 2+2=4.

Demonstrate yes. Prove no. There is no empirical -proof-. Let a, b, c, d be distinct objects. No pair are indistinguishable. Now consider the sets {a, b} and {c,d}. The first set has two objects. The second set has two objects. Take the union and get {a,b,c,d}. Since all the objects are distinct (by assumption) this last set has 4 (count'em) objects. This is essentially how we teach children then two and two make four.

To -prove- that 2 and 2 make 4 you need an axiomatic basis for the arithmetic of integers. The Peano Axioms (Google) is such a basis.

Ba'al Chatzaf

[tjohnson]

Demonstrate yes. Prove no. There is no empirical -proof-. Let a, b, c, d be distinct objects. No pair are indistinguishable. Now consider the sets {a, b} and {c,d}. The first set has two objects. The second set has two objects. Take the union and get {a,b,c,d}. Since all the objects are distinct (by assumption) this last set has 4 (count'em) objects. This is essentially how we teach children then two and two make four.

To -prove- that 2 and 2 make 4 you need an axiomatic basis for the arithmetic of integers. The Peano Axioms (Google) is such a basis.

Ba'al Chatzaf

2 objects combined with 2 objects gives you 4 objects yes, this is applied mathematics. This forum has numerous postings by people who seem to have no knowledge of the difference between mathematics and applied mathematics.

[Michael Stuart Kelly]

GS,

In Objectivism, all symbolic systems like words, logic and math have their basis in reality. Math is abstract, but it does follow the law of identity. As reality does also, it works. That's the initial connection. It is not just a set of arbitrary rules in the mind completely disconnected from the rest of reality.

I connected one set of dots and came up with the following. We are part of reality, therefore we fall under the same laws of organization that govern reality, therefore, our mental system of organization (even when using symbols) works according to the laws of organization that govern reality.

So in a metaphysical sense (law of identity), all math is applied math. A math unit is a "thing." It is just a mental thing and not a perceptual thing.

Michael

[BaalChatzaf]

GS,

In Objectivism, all symbolic systems like words, logic and math have their basis in reality. Math is abstract, but it does follow the law of identity.

The law of non-contradiction. Mathematics done abstractly and formally need have no empirical content whatsoever. As long as there is one well formed expression in a mathematical system that cannot be derived from its postulates, all is well.

Ba'al Chatzaf

[tjohnson]

GS,

In Objectivism, all symbolic systems like words, logic and math have their basis in reality. Math is abstract, but it does follow the law of identity. As reality does also, it works. That's the initial connection. It is not just a set of arbitrary rules in the mind completely disconnected from the rest of reality.

I connected one set of dots and came up with the following. We are part of reality, therefore we fall under the same laws of organization that govern reality, therefore, our mental system of organization (even when using symbols) works according to the laws of organization that govern reality.

So in a metaphysical sense (law of identity), all math is applied math. A math unit is a "thing." It is just a mental thing and not a perceptual thing.

Michael

I understand very little of what you are saying here and can only tell you that in General Semantics mathematics and applied mathematics are considered two very different things and from reading the interminable posts showing the confusion between the two in this forum I would highly recommend trying a different approach.

Edited September 12, 2007 by general semanticist

[Michael Stuart Kelly]

Bob,

But it must have a concrete form, at least one that can be imagined, and that concrete form must follow the law of identity (which is grounds for the law of non-contradiction). Otherwise it cannot be used.

A number, for example, is one such form.

There's your empirical connection.

Michael

[BaalChatzaf]

Bob,

But it must have a concrete form, at least one that can be imagined, and that concrete form must follow the law of identity (which is grounds for the law of non-contradiction). Otherwise it cannot be used.

A number, for example, is one such form.

There's your empirical connection.

Michael

Can you come up with a concrete form for a space with an uncountably infinite number of dimensions? I can't. But I can come up with a formal system that deals with such spaces.

Please do not confuse the description with the thing described.

Ba'al Chatzaf

[Michael Stuart Kelly]

What does that got to do with it?

Your so-called space is merely a logical extension of the concretes. Without the concretes, it would not be imagined at all.

Michael