Skeptics "reasoning" applied to mathematics

[primemover]

Please forgive the lapse but I can't resist. Are you absolutely 100% certain about what you are proposing? You seem emotionally committed to it.

You seem to see a bit too often all kinds of emotions in other people, do you think it would invalidate their arguments, while you have no emotional commitments at all?

In this case my statement is just the advice to use a sensible definition of "knowledge". You shouldn't take words like "certain" out of their context, which is the common error in this kind of (hardly original) questions.

Your definition of knowledge seems to be whatever one thinks, period.

[Dragonfly]

So you would agree with Rand about contextual certainty, then?

No, because she meant something entirely different by that.

[Daniel Barnes]

Michael:

>I do remember typing a friendly post on your blog. I might have been mistaken, though.

My very point.

[primemover]

We don't need 100% certainty to know something.

When I am not certain about something I am careful not to say I know such is the case and then I may state what I think a good probability of what the case might be . But I never claim that I know something and simultaneously not sure about it.

That absolute knowledge about the world is not possible does not imply that no knowledge is possible.

Again with this wishy washy definition of knowledge to be whatever one thinks.

A sufficiently high probability is in practice good enough. Knowledge is that what makes our living possible, even if it is not 100% certain. Your error is to equate knowledge with 100% certainty, implying that everything you "know" is 100% certain and that you never err. Yeah, sure.

No , in that if I find out I was wrong about X, then I am honest and say "well damn I guess I didn't know about how X functions after all." I wouldn't still claim to know how X functions. Bu then again, I would have said that I was under the impression that X functioned that way.

Edited April 30, 2007 by primemover

[Dragonfly]

Your definition of knowledge seems to be whatever one thinks, period.

Not whatever one thinks. Some propositions will have more evidence for them or be more likely than other ones and it is a good idea to realize the difference.

[Michael Stuart Kelly]

So you would agree with Rand about contextual certainty, then?

No, because she meant something entirely different by that.

Dragonfly,

As you know, I don't mind the same word meaning more than one concept. It just needs to be clarified when there is confusion.

This is one such case. Could you please explain your meaning as opposed to Rand's? I don't know what you mean in this case.

Michael

[primemover]

Your definition of knowledge seems to be whatever one thinks, period.

Not whatever one thinks. Some propositions will have more evidence for them or be more likely than other ones and it is a good idea to realize the difference.

Ok so you agree that there is a method for determining what constitutes knowledge and what doesn't?

[Victor Pross]

Primemover,

I am theoretically taking leave of posting, but damn it, I saw this thread and immediately identified and sympathized with your position.

Generally speaking, I find skepticism to be an unintelligent and silly—not to mention self-canceling--philosophical position to take. When at college taking philosophy, I routinely fell into debates with skeptics and this thread brought back many memories.

Prime, it all starts with Hume. I don’t think we would be having this conversation were it not for Hume’s obtuse “contributions” to philosophy. Hume’s philosophy, whether true or false, epitomizes the bankruptcy of eighteen-century romanticism. Hume starts out, as did Locke, with the aim of being sensible and empirical, taking nothing “on trust,” but seeking whatever instruction is to be taken from ‘experience’ and ‘observation.’ But then Hume arrives at the catastrophic conclusion that from experience and observation nothing is to be learnt!*(1.1)

Sidebar: Of course, Daniel and Dragonfly are trying to “soften” skepticism, trying to make it sound less ridiculous than it really is. For Hume, however, there is no such thing as a rational belief: “If we believe that fire warms or that water refreshes, ‘tis only because it costs us too much pains to think otherwise.” You see, we cannot help believing, but no belief can be grounded in reason.*(1.2) This is Hume’s stance. And this anti-rationality is nonsense. In a certain way, Hume’s skepticism is insincere, since he could not maintain it in practice.

Take note that skepticism as a philosophy is not about 'doubt' per se or about “reasonable doubt” on dubious propositions—such as “God exists”—but rather: it is about dogmatic doubt. The honest man of science says: “I think it is such-and-such, but I am not sure.” The man of intellectual honesty says: “I don’t know how it is, but I hope to find out.” The philosophical skeptic—thinking himself so superior—says: “Nobody knows, and nobody can ever know.”*(1.3) (And this is considered to be "wisdom").

Skeptics, being the back-peddlers they are, deny that they assert the impossibility of knowledge, but their denials are not very convincing when you are the “philosophical sons” of David Hume. :turned:

-Victor

NOTE FROM ADMINISTRATOR:

*(1) Plagiarized from A History of Western Philosophy by Bertrand Russell. The original passage reads as follows:

(1.1) (p. 672)

Hume's philosophy, whether true or false, represents the bankruptcy of eighteenth-century reasonableness. He starts out, like Locke, with the intention of being sensible and empirical, taking nothing on trust, but seeking whatever instruction is to be obtained from experience and observation. But having a better intellect than Locke's, a greater acuteness in analysis, and a smaller capacity for accepting comfortable inconsistencies, he arrives at the disastrous conclusion that from experience and observation nothing is to be learnt.

(1.2) (p. 672)

There is no such thing as a rational belief: "If we believe that fire warms, or water refreshes, 'tis only because it costs us too much pains to think otherwise." We cannot help believing, but no belief can be grounded in reason.

(1.3) (p. 233)

It should be observed that Scepticism as a philosophy is not merely doubt, but what may be called dogmatic doubt. The man of science says "I think it is so-and-so, but I am not sure." The man of intellectual curiosity says "I don't know how it is, but I hope to find out." The philosophical Sceptic says "nobody knows, and nobody ever can know."

OL extends its deepest apologies to the heirs of Bertrand Russell.

Edited August 1, 2007 by Michael Stuart Kelly

[Daniel Barnes]

Victor:

>Skeptics, being the back-peddlers they are...

Hey, at least we're not serial plagiarists!

[Victor Pross]

Primemover,

But I am curious to ask now: have you ever read Hume or other skeptic's works? Do you have any ideas what would make this philosophical orientation compelling to certain intellectuals?

-Victor

Edited May 1, 2007 by Victor Pross

[Dragonfly]

Victor:

>Skeptics, being the back-peddlers they are...

Hey, at least we're not serial plagiarists!

Note he's also talking about the "man of intellectual honesty", as if he knows what that means...

[Daniel Barnes]

Dragonfly:

>Note he's also talking about the "man of intellectual honesty", as if he knows what that means...

Ah, but my dear Dragonfly, I am curious to ask now: have you ever pretended to have read Hume or other skeptic's (sic) works on certain internet fora, when you really hadn't?

Do you have any ideas what would make such pretences - not to mention the blatant and repeated pinching other people's writing - compelling to certain wannabe-intellectuals?

[BaalChatzaf]

description. Of mathematical statements we can be certain, as they are derived from the definitions; of statements about the real world we can never be 100% certain, as we never can learn all the variables and we never can be sure that our model is correct. That is in a nutshell the difference between analytical statements and synthetic statements. Welcome to the real world.

Maybe yes, maybe no. For example Wile's proof of Fermat's Last Theorem (so-called) is so complex that it had to be vetted by a committee of top notch number theorists and group theorists. Wile's first version of the proof was found to be defective but so far no one has found a defect in his latest proof. But when a proof runs hundreds of pages in length there is always a chance something is being overlooked. So I would say it is highly likely, but not absolutely certain that FLT has been proven to be true.

Ba'al Chatzaf

[BaalChatzaf]

When I am not certain about something I am careful not to say I know such is the case and then I may state what I think a good probability of what the case might be . But I never claim that I know something and simultaneously not sure about it.

Or you might modestly assert that you believe such and such to be the case and present the reasons for your belief. The things we -know- are mostly what we believe to be the case or accept to be the case. We may have good reasons for our belief. Or maybe we have no good reason to believe the opposite.

In any case, the only things you truly KNOW are those you know first hand and by your own witness or experience. Anything else is a kind of hearsay. Most of what we say we know, is really received from others. This is not as bad as it sounds. If we were restricted strictly to what we know at first hand, we would know damned little and we would have no access to the experience of others. It would be as if we were cut loose in the world as soon as we could physically survive by ourselves. We would be in the position of hermits on a desert island. Our lives would be nasty brutish and short. Living in society and mostly trusting in the word and judgment of our fellows gives us a reach into the world we otherwise would not have.

Ba'al Chatzaf

[Dragonfly]

Maybe yes, maybe no. For example Wile's proof of Fermat's Last Theorem (so-called) is so complex that it had to be vetted by a committee of top notch number theorists and group theorists. Wile's first version of the proof was found to be defective but so far no one has found a defect in his latest proof. But when a proof runs hundreds of pages in length there is always a chance something is being overlooked. So I would say it is highly likely, but not absolutely certain that FLT has been proven to be true.

That's true (another example would be the proof of the four color theorem), but that is more a practical problem due to the complexity and the long chain of reasoning. In principle certainty is here possible, if not in (current) practice.

[Brant Gaede]

Maybe yes, maybe no. For example Wile's proof of Fermat's Last Theorem (so-called) is so complex that it had to be vetted by a committee of top notch number theorists and group theorists. Wile's first version of the proof was found to be defective but so far no one has found a defect in his latest proof. But when a proof runs hundreds of pages in length there is always a chance something is being overlooked. So I would say it is highly likely, but not absolutely certain that FLT has been proven to be true.

That's true (another example would be the proof of the four color theorem), but that is more a practical problem due to the complexity and the long chain of reasoning. In principle certainty is here possible, if not in (current) practice.

If Fermat had a proof of his theorem, as he claimed, it didn't take hundreds of pages. So if a shorter proof could be found, that would be interesting to say the least.

--Brant

[Dragonfly]

This is one such case. Could you please explain your meaning as opposed to Rand's? I don't know what you mean in this case.

First, as far as I know this is not Rand's formulation, but a concoction by Peikoff. His idea is that if you arrive at a conclusion by logical reasoning using all the information that is available at that moment, you can be "certain" of that conclusion, it is the "truth". Now even Peikoff knows that it is possible that later more information becomes available that falsifies this "truth", so while he calls it the truth or a certainty, he uses the qualifier "contextual". So in his view the notions that for example the influence of gravity is instantaneous and that time is absolute are "contextual truths", which is just a dirty euphemism for "proven false" and "not certain", but these terms sound so unobjectivistic.

When I use the term "context" it refers to the object of the certainty statement; where we cannot be 100% certain is in general positive statements about the physical world, in other words, in the results, the laws of the sciences. There are other contexts where this is not relevant, as in logical or mathematical reasoning or in certain meta-statements. Further there is no point in doubting the negation of obviously absurd statements, like the idea that I am a teapot. So merely jumping at every appearance of the word "certain" is not very constructive neither original, as you have to consider the context in which it is used. That also applies to the context of Peikoff's statements about "contextual certainty" (2 different meanings of "context" here!), which is just a weasel term for "less than 100% certainty". In fact Peikoff admits that 100% certainty about the physical world is not possible, but he hides it behind a terminological smoke screen. Our objection is against the weaseliness and the confusing power of the term.

Anyway, I hope my explanation of the difference is clear now to you.

[Dragonfly]

If Fermat had a proof of his theorem, as he claimed, it didn't take hundreds of pages. So if a shorter proof could be found, that would be interesting to say the least.

No doubt that would be interesting, and perhaps a shorter proof may be found in the future, as the result of new developments in mathematics.

But the idea that Fermat did have a correct proof is extremely unlikely, as the mathematical methods and theories that were known at his time were just not up to this task. He was probably bluffing, joking or had a fallacious proof.

[Dragonfly]

Ok so you agree that there is a method for determining what constitutes knowledge and what doesn't?

The scientific method works fine.

[Brant Gaede]

If Fermat had a proof of his theorem, as he claimed, it didn't take hundreds of pages. So if a shorter proof could be found, that would be interesting to say the least.

No doubt that would be interesting, and perhaps a shorter proof may be found in the future, as the result of new developments in mathematics.

But the idea that Fermat did have a correct proof is extremely unlikely, as the mathematical methods and theories that were known at his time were just not up to this task. He was probably bluffing, joking or had a fallacious proof.

If he didn't have a proof or didn't think he had then he was putting forth a challenge for math guys to find one and/or giving himself unearned credit. Yeah, joking too, maybe. But it was a lot more interesting the way he did it than any other way I can imagine, other than stating he had buried it in a bottle somewhere, go have a treasure hunt.

--Brant

[primemover]

First, as far as I know this is not Rand's formulation, but a concoction by Peikoff. His idea is that if you arrive at a conclusion by logical reasoning using all the information that is available at that moment, you can be "certain" of that conclusion, it is the "truth". Now even Peikoff knows that it is possible that later more information becomes available that falsifies this "truth",

No that is not what Peikoff says. He says that if you form your concepts properly, new information adds to the previous information.

so while he calls it the truth or a certainty, he uses the qualifier "contextual". So in his view the notions that for example the influence of gravity is instantaneous and that time is absolute are "contextual truths", which is just a dirty euphemism for "proven false" and "not certain", but these terms sound so unobjectivistic.

No, a new variable for time ( how fast your are moving through time) is what was discovered. Newtons work was very important for expanding on this knowledge.

When I use the term "context" it refers to the object of the certainty statement; where we cannot be 100% certain is in general positive statements about the physical world, in other words, in the results, the laws of the sciences. There are other contexts where this is not relevant, as in logical or mathematical reasoning or in certain meta-statements. Further there is no point in doubting the negation of obviously absurd statements, like the idea that I am a teapot. So merely jumping at every appearance of the word "certain" is not very constructive neither original, as you have to consider the context in which it is used. That also applies to the context of Peikoff's statements about "contextual certainty" (2 different meanings of "context" here!), which is just a weasel term for "less than 100% certainty". In fact Peikoff admits that 100% certainty about the physical world is not possible, but he hides it behind a terminological smoke screen. Our objection is against the weaseliness and the confusing power of the term.

Anyway, I hope my explanation of the difference is clear now to you.

I could have swore he was asking you for your definition of "knowledge".

[primemover]

Ok so you agree that there is a method for determining what constitutes knowledge and what doesn't?

The scientific method works fine.

And why do you think that is? Why isn't it that majority consensus or authority validates truths? The very foundation for the scientific methods rest on the very idea that A is A and that existence has primacy over consciousness.

Edited May 3, 2007 by primemover

[primemover]

Worthy of quotation from Victor.

Take note that skepticism as a philosophy is not about 'doubt' per se or about “reasonable doubt” on dubious propositions—such as “God exists”—but rather: it is about dogmatic doubt. The honest man of science says: “I think it is such-and-such, but I am not sure.” The man of intellectual honesty says: “I don’t know how it is, but I hope to find out.” The philosophical skeptic—thinking himself so superior—says: “Nobody knows, and nobody can ever know.” (And this is considered to be "wisdom").

[primemover]

Primemover,

But I am curious to ask now: have you ever read Hume or other skeptic's works? Do you have any ideas what would make this philosophical orientation compelling to certain intellectuals?

-Victor

Yes, Hume and Descartes mainly.

What draws people to it? Perhaps because it is emotionally satisfying in a way that a theist is emotional satisfied by thinking a magic man in the sky is watching them.

[Guyau]

Here is David Hume's argument to general skepticism:

In every judgment, which we can form concerning probability, as well as concerning knowledge, we ought always to correct the first judgment, deriv'd from the nature of the object, by another judgment, deriv'd from the nature of the understanding. . . . Our sentiments have different degrees of authority . . . in proportion to the degrees of our reason and experience. In the man of the best sense and longest experience, this authority is never entire; since even such-a-one must be conscious of many errors in the past, and must still dread the like for the future. Here then arises a new species of probability to correct and regulate the first, and fix its just standard and proportion. As demonstration is subject to the controul of probability, so is probability liable to a new correction by a reflex act of mind, wherein the nature of our understanding, and our reasoning from the first probability become our objects.

Having thus found in every probability, beside the original uncertainty inherent in the subject, a new uncertainty deriv'd from the weakness of that faculty . . . .

Let our first belief be ever so strong, it must infallibly persist by passing thro' so many new examinations, of which each diminishes somewhat of its force and vigour. . . . And when I proceed still farther, to turn the scrutiny against every successive estimation I make of my faculties, all the rules of logic require a continual diminution and at last a total extinction of belief and evidence.

Treatise 1.4.1

Against such an outcome of general skepticism in antiquity, we find the Epicurean Lucretius writing:

If anyone thinks that nothing is known, he does not even know whether that can be known, since he declares that he knows nothing. I will therefore spare to plead cause against a man who has placed his head in his own footsteps. And yet even if I grant that he knows that, still I will ask just this: since material things had no truth for his vision to begin with, how he knows what it is to know or not to know as the case may be, what gave him the concept of true and false, what evidence proved that the doubtful differs from the certain.

You will find that it is from the senses in the first instance that the concept of truth has come, and that the senses cannot be refuted. For some standard must be found of greater credit, able of itself to refute false things by true. What, moreover, must be held of greater credit than the senses? Shall reasoning, derived from false sense, prevail against these senses, being itself wholly derived from the senses?

On the Nature of Things 4.469-87

Do readers here think that the Epicurean argument is sufficient to defeat Hume's argument?

What further failings of this particular Humean argument are exposed by Rand's metaphysics and epistemology, beyond the failure signaled by Lucretius?

Edited May 3, 2007 by Stephen Boydstun