## February 27, 2005

### WE ARE ALL PLATONISTS NOW:

TIME BANDITS: What were Einstein and Gödel talking about? (JIM HOLT, 2005-02-28, The New Yorker)

Gödel entered the University of Vienna in 1924. He had intended to study physics, but he was soon seduced by the beauties of mathematics, and especially by the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind. This doctrine, which is called Platonism, because it descends from Plato’s theory of ideas, has always been popular among mathematicians. In the philosophical world of nineteen-twenties Vienna, however, it was considered distinctly old-fashioned. Among the many intellectual movements that flourished in the city’s rich café culture, one of the most prominent was the Vienna Circle, a group of thinkers united in their belief that philosophy must be cleansed of metaphysics and made over in the image of science. Under the influence of Ludwig Wittgenstein, their reluctant guru, the members of the Vienna Circle regarded mathematics as a game played with symbols, a more intricate version of chess. What made a proposition like “2 + 2 = 4” true, they held, was not that it correctly described some abstract world of numbers but that it could be derived in a logical system according to certain rules.Gödel was introduced into the Vienna Circle by one of his professors, but he kept quiet about his Platonist views. Being both rigorous and averse to controversy, he did not like to argue his convictions unless he had an airtight way of demonstrating that they were valid. But how could one demonstrate that mathematics could not be reduced to the artifices of logic? Gödel’s strategy—one of “heart-stopping beauty,” as Goldstein justly observes—was to use logic against itself. Beginning with a logical system for mathematics, one presumed to be free of contradictions, he invented an ingenious scheme that allowed the formulas in it to engage in a sort of double speak. A formula that said something about numbers could also, in this scheme, be interpreted as saying something about other formulas and how they were logically related to one another. In fact, as Gödel showed, a numerical formula could even be made to say something about itself. (Goldstein compares this to a play in which the characters are also actors in a play within the play; if the playwright is sufficiently clever, the lines the actors speak in the play within the play can be interpreted as having a “real life” meaning in the play proper.) Having painstakingly built this apparatus of mathematical self-reference, Gödel came up with an astonishing twist: he produced a formula that, while ostensibly saying something about numbers, also says, “I am not provable.” At first, this looks like a paradox, recalling as it does the proverbial Cretan who announces, “All Cretans are liars.” But Gödel’s self-referential formula comments on its provability, not on its truthfulness. Could it be lying? No, because if it were, that would mean it could be proved, which would make it true. So, in asserting that it cannot be proved, it has to be telling the truth. But the truth of this proposition can be seen only from outside the logical system. Inside the system, it is neither provable nor disprovable. The system, then, is incomplete. The conclusion—that no logical system can capture all the truths of mathematics—is known as the first incompleteness theorem. Gödel also proved that no logical system for mathematics could, by its own devices, be shown to be free from inconsistency, a result known as the second incompleteness theorem.

Wittgenstein once averred that “there can never be surprises in logic.” But Gödel’s incompleteness theorems did come as a surprise. In fact, when the fledgling logician presented them at a conference in the German city of Königsberg in 1930, almost no one was able to make any sense of them. What could it mean to say that a mathematical proposition was true if there was no possibility of proving it? The very idea seemed absurd. Even the once great logician Bertrand Russell was baffled; he seems to have been under the misapprehension that Gödel had detected an inconsistency in mathematics. “Are we to think that 2 + 2 is not 4, but 4.001?” Russell asked decades later in dismay, adding that he was “glad [he] was no longer working at mathematical logic.” As the significance of Gödel’s theorems began to sink in, words like “debacle,” “catastrophe,” and “nightmare” were bandied about. It had been an article of faith that, armed with logic, mathematicians could in principle resolve any conundrum at all—that in mathematics, as it had been famously declared, there was no ignorabimus. Gödel’s theorems seemed to have shattered this ideal of complete knowledge.

That was not the way Gödel saw it. He believed he had shown that mathematics has a robust reality that transcends any system of logic. But logic, he was convinced, is not the only route to knowledge of this reality; we also have something like an extrasensory perception of it, which he called “mathematical intuition.” It is this faculty of intuition that allows us to see, for example, that the formula saying “I am not provable” must be true, even though it defies proof within the system where it lives. Some thinkers (like the physicist Roger Penrose) have taken this theme further, maintaining that Gödel’s incompleteness theorems have profound implications for the nature of the human mind. Our mental powers, it is argued, must outstrip those of any computer, since a computer is just a logical system running on hardware, and our minds can arrive at truths that are beyond the reach of a logical system.

The Materialists still take it rather poorly when you point out that their Reason and Logic rest on a basis of Faith (or intuition as Godel would have it). But Godel was correct, it simply means that our access to ultimate reality is not rational, but faith-based. Posted by Orrin Judd at February 27, 2005 10:24 AM

Oh dear. This is worse than the abuse of the uncertainty principle.

Orrin, please take an elementary course in logic and formal languages and spare yourself further embarrassment.

Posted by: Dutch at February 27, 2005 3:31 PMOJ: is not his "mathematical intuition" better likened to natural law?

Posted by: Palmcroft at February 27, 2005 3:39 PMIf this sort of thing facinates. Damage your brain further with a Classical Mechanics course.

Posted by: Tom Wall at February 27, 2005 4:34 PMThe way I heard it, all Godel showed is that not all statements expressable within a system are provable within that system. Nothing about their being true or false. In some cases, you can even create new, valid systems, based on the possible values of the unprovable statemenets.

The classic example is the Parallel Postulate, which is independent of the other Postulates, and has three possible values (As in the math joke that says that in math there are only three number: zero, one and many.) It was recognition of this that gave rise to non-Euclidian geometry. All three of these are imbedded in a unified geometry which itself has unprovable statements which may spawn even more, larger, more complex systems. No matter how big you get, you never can be "complete", which was Russell's great goal, and why he figures in all this.

So where does the leap from this simple statement of logical systems to "intuition" come from?

Posted by: Raoul Ortega at February 27, 2005 5:13 PMAnd those systems will have the same problems too, ad infinitum...

Posted by: oj at February 27, 2005 5:20 PMDutch:

There's no reason to be embarrassed that logic isn't based on logic.

Ever read Heisenberg?

http://www.brothersjudd.com/blog/archives/014309.html

Posted by: oj at February 27, 2005 5:26 PMOJ:

A logic is a system of rules which governs the assignment of truth values. Why should it "be based on logic"? What does that even mean?

Posted by: Dutch at February 27, 2005 7:08 PMDutch:

Yes, the rules can not be derived logically, they have to be taken on faith--or authority, if you find that less threatening.

Posted by: oj at February 27, 2005 7:27 PMOJ:

No, they don't have to be taken on faith. Either the rules have properties which satisfy one's purpose or they don't. In the latter case you pick a set of rules which does. But this is a specialist's concern. It doesn't really provide grounds for the kind of sweeping generalizations you are making.

Posted by: at February 27, 2005 7:47 PMThere is no logical basis for the rules, thus every logical system depends on illogic.

Posted by: oj at February 27, 2005 8:13 PMOJ:

Perhaps (I suppose it depends on what you mean), but that doesn't mean that they don't have a rational basis, and it also doesn't have anything to do with Godel's incompleteness proof.

Posted by: Dutch at February 27, 2005 8:26 PMDutch:

That's precisely the point, there's no basis for Reason. However, there was some comfort in the idea that if, with Descartes, we just accept that we exist and that reason is possible on the basis of faith, that Reason, Logic, Math, Science,m etc. would prove internally consistent, complete, and provide stable and coherent depictions of "reality." Heisenberg, Schroedinger, Godel, etc. showed they don't even succeed at that. There's no comfort to be had other than faith. But, of course, modern secular materialists can't acknowledge that.

Posted by: oj at February 27, 2005 9:33 PMNo, oj, we're not all Platonists now. Even in the wake of Goedel nobody went back to believing that the natural numbers were some sort of mysterious essences. Numbers are sets of sets now, as they have been ever since Russell and Whitehead got done with them.

Posted by: joe shropshire at February 27, 2005 9:52 PMI don't have a lot of time, so I'll be brief, Orrin.

There is no dispute here that there are true mathematical statements and false mathematical statements.

Proofs are methods whereby, given known true mathematical statements, one can get new mathematical statements that are also true if the methods are applied correctly.

A logical system is a set of mathematical statements that are held to be true (axioms), conjoined with a limited set of methods for combining true statements to get other statements.

A proof is a set of true mathematical statements combined by the allowable methods that yields new statements.

A logical system is **inconsistent** if, in the process of generating proofs, one chain of proofs leads to a specific statement, while a different chain of proofs leads to that specific statement's logical negation.

The Vienna circle was motivated by the prospect of figuring out a limited set of methods and a mechanical means for applying them to a basic set of axioms so that mathematical truths could be generated like numbers from a calculator.

The main problem facing the Vienna Circle was the question of whether there were mathematical truths out there that NEEDED to be axioms, because it was recognized, from the work on Non-euclidean geometries, that there were multiple sets of axioms that could lead to internally consistent (non-contradictory) mathematical systems, but which could NOT be combined willy-nilly, since the axioms that differentiated the non-euclidean geometries CONTRADICTED each other. This is the essence of Raoul Ortega's comment on the parallel postulate.

Thus, they already knew that if one added too many axioms, the system could be contradictory. However, if one started off with too few, there were truths that could not be cranked out (proved), because the axiom left out proved to be the only "bridge"/"gateway" to those truths. Such a system with too few axioms was "incomplete", in that there were (acknowledged) truths out there, but which couldn't be mechanistically "generated" (proved) by such a system.

Russel and whitehead's principa mathematica was an attempt to come up with a core set of axioms that would lead to a "complete" system: that is, one with a set of axioms and methods of proof that would be guaranteed to generate ALL truths. Thus, all truths would be provable.

They only published a volume or two, I believe, and stopped because Godel's incompleteness theorem showed that, given THEIR starting point, there was at least one truth out there that was not provable, because he was able to use the axioms and methods to construct (mechanistically) that one truth, which essentially says "Given these axioms and methods of derivation, this statement cannot be proved."

Whitehead and Russell's "fix", upon finding such a truth that wasn't provable, was to include it as an axiom (I.e. patch Version X to Version X'). Godel's proof showed how, even if you included the unprovable, but true, statement as an axiom, the resulting system could be made to construct a NEW version of the statement "Given this expanded set of axioms and methods of derivation, this (new) statement cannot be proved." (Relative to Version X'.)

Thus, in response to a movement in mathematics that tried to come up with a mechanical means to prove all mathematical truth for all time, in essence constructing a Great Mathematical Truth Generation Machine, Godel came up with a mechanical means to demonstrate that there was a mathematical truth that could not be proved, no matter what set of starting conditions was given to the Great Mathematical Truth Generation Machine. I.e. No matter what version Z of the machine the circle could come up with, Godel could prove that Z wasn't good enough by producing a mathematical truth that was obviously true, but could also be PROVED that Z COULDN'T prove it.

"Illogical" is probably the wrong word to use: Axiomatic or fundamental or non-derivable would be better, and more appropriate, terms. There is nothing "illogical" about the kinds of truths Godel's method produces. it's just those truths ALSO say two things about themselves: "We are true", and "We can't be proved." The first is a statement of the nature of those truths, and is supposed to be self-evident. The second is NOT a statement about the truth, but about the capability of a specific mathematical system to PROVE that it is true.

joe:

You have that exactly backwards. After Godel we all went back, as Ptah above, to believing in essences, accepting our faith as sufficient.

Posted by: oj at February 27, 2005 10:48 PMDutch:

They aren't windmills, they're giants. No one liked the windmill world:

http://www.brothersjudd.com/blog/archives/019646.html

Posted by: oj at February 27, 2005 10:49 PM"What we demand are rigidly defined areas of doubt and uncertainty."

- Vroomfondel, the representative of the Amalgamated Union of Philosophers, Sages, Luminaries, and Other Professional Thinking Persons.

Posted by: Gideon at February 27, 2005 11:58 PMNot so, oj, at least so far as arithmetic is concerned. The idea that the *Principia* proposes for the nature of, say, the number three is that it's merely a common property (the cardinality) of all sets that count out at three members. In other words, the number three doesn't live in some mysterious parallel ideal world, as Plato thought; rather it's grounded in the prosaic act of *counting*: a-one, and a-two, and a-three. That's about as far from Platonism as one can get, and Goedel's results don't reverse that. Cardinal arithmetic as developed by Russell and Whitehead is one example of what the honest use of abstraction looks like. They really did labor to develop tools that could grasp the concrete without squeezing it out of all shape; and that's all we can ask of our philosophers. I know you're suspicious of pure abstraction wherever you find it, but in fact it's just another medium, like wood or stone, that an artisan can handle well or poorly.

joe:

No one truly believes that there's a place where counting one, two, three renders four, but logically there's no reason you couldn't make that the rule.

Posted by: oj at February 28, 2005 1:43 AMI think we can summarise OJ's position thus (please correct me if I'm wrong, OJ):

He accepts that 'Reason' (including logic, maths, the scientific method, evidence etc) can be consistent or inconsistent within its own rules and boundaries.

But, he argues, when you make a decision or state a proposition based on 'Reason', you are making a 'leap of faith' that your decision or proposition, because it is consistent within these rules, is superior or better-founded than any old statement based on Faith.

Well, in a way he's right - you are. But that's exactly the difference between Reason and Faith. Tell us something we don't know.

By redefining 'Faith' to include 'Reason' OJ is not making an insight, he's just destroying the standard semantic definitions.

This redefinition only allows you to say: "therefore a statement based on Reason is identical to a statement based on Faith, according to my new definitions'.

Even if this were valid, it makes OJ's preference for Faith OVER Reason meaningless, since they're the same thing.

It certainly can't get you to 'Faith is superior to Reason', which he also wants to say.

Posted by: Brit at February 28, 2005 6:41 AMBrit:

You, as many, misunderstand the respective truth claims of Reason and Faith and therefore fail to understand how devastating the recognition that Reason is based on faith was.

Posted by: oj at February 28, 2005 7:53 AMYes, so you keep saying.

Since I'm so ignorant, how about explaining to me, in words that I can understand, why this is the case and what those devastating implications are.

Because I keep reading your statements and I'm none the wiser.

Or is it a secret?

Posted by: Brit at February 28, 2005 8:17 AMBrit:

Not secret, just rational so hard for the faithful to accept.

The premise of Faith is that we have access to the truth by faith alone, though we can use various faith-based tools to access it, like reason.

The premise of Reason is that we can only call something true if it is supported by reason, that faith does not provide a sufficient base for truth claims. However, since reason itself ultimately rests on a series of faith claims reason destroys the pretense of Reason itself.

This was recognized even by the Greeks, but demonstrated most significantly by Hume, which is how the Anglo-American world avoided much of the dead end of the Age of Reason. However, those who raced down that blind alley were in for such a rude awakening when they hit the wall that they (you) have still not recovered.

OJ:

Thanks.

Suppose youre going to put a pound of sugar in one side of a set of scales, and a half-pound in the other. Which side will be lower? All the faith in the world wont mean that the scales will balance, or the half-pound side will be lower. Reason allows you to predict that the pound side will be lower, and you could construct a Reason-based proof that it will. Thats a question at a nice practical level.

Uncontroversial so far?

Now, when it comes to certain deep epistemology-level questions, such as: "does objective reality exist, or might I just be a brain in a vat?" then its true that Reason can never allow us to prove that Im not just a brain in a vat.

(Though Reason can tell you: its true that I cant prove Im not a brain in a vat, but I dont have any reason to believe that I am)

Therefore, for all questions above this deep level, we have to implicitly assume that objective reality exists, that Im not a brain in a vat, etc.

This is the case whether you value Faith or Reason most highly.

(You can call these implicit assumptions faiths if you like, so long as you dont confuse it with Faith.)

Now, do these assumptions (or faiths) mean that Reason the approach that says that you cannot call something definitely true without a basis in reason is undermined when it comes to matters above this deep level, such as balancing scales or Newtonian physics, or who will win the Superbowl or evolution or whether God exists?

You say yes, I say no.

Reason is still a much more useful tool than Faith, because it allows you to choose between competing theories, while Faith does not.

A mistake you might make is to think that questions like does God exist? or is evolution the result of natural processes are at the deep epistemology level.

Theyre not theyre at a practical enough level so that Reason can allow you to decide between the competing theories, and you can pinpoint the areas where the best answer is we dont know yet.

Brit:

No, already controversial. You can't get to the point of sugar by the exercise of reason.

Posted by: oj at February 28, 2005 10:51 AMSo what? If you read the rest of the post you'll note that I agree that we must make certain assumptions about the existence of objective reality, whether applying Reason or Faith.

Posted by: Brit at February 28, 2005 11:09 AM"The premise of Faith is that we have access to the truth by faith alone, though we can use various faith-based tools to access it, like reason."

Which faith?

YECs and OECs and IDers all make competing, very faith based, claims about the same objectve reality--whatever it might be.

They can't all be correct at the same time.

So which one is correct? How did you decide? If you decided all are incorrect, how did you do that?

"The premise of Reason is that we can only call something true if it is supported by reason, that faith does not provide a sufficient base for truth claims."

Strawman alert. The premise of Reason is that, within severe constraints, we can judge the relative truth value of competing explanations. Outside those constraints, such discrimination is impossible.

That's it. Perhaps your blood pressure would go down if you started viewing reason as it it, rather than some Monsters Inc boogeyman.

Posted by: Jeff Guinn at February 28, 2005 11:48 AMJeff:

As does Darwinism. God left us with Free Will to decide among the competing claims but told us which is right.

Posted by: oj at February 28, 2005 11:56 AMa) The premises are the same as yours. The conclusion is different.

b) Because I took the trouble to write it and it would only be polite.

Brit:

No, your premise is nonsense and so must everything that follows be. Until you can prove via reason that you exist--something no one has ever succeeded at--you can't begin to discuss Reason rationally. If you want to discuss it based on faith we can.

Posted by: oj at February 28, 2005 12:09 PMThat's what you said before I made my objection. If you can't address my objection, fine, but we're not arguing. You're just ranting.

Posted by: Brit at February 28, 2005 12:14 PMBrit:

What objection? Prove just "I" and we can move on. No one ever has.

Posted by: oj at February 28, 2005 12:22 PMYour argument runs:

Premise: Whether using Faith or Reason, we have to make certain assumptions at the deepest philosophical levels which Reason cannot prove, eg. "I exist".

Conclusion: Therefore Reason is undermined and is not useful for theories above this level.

My argument runs:

Premise: Whether using Faith or Reason, we have to make certain assumptions which Reason cannot prove, eg. "I exist".

Conclusion: However, Reason is still a valid and useful tool for deciding between competing theories above this level.

You haven't shown why your conclusion follows the premise.

I've demonstrated that mine appears to follow from the premise with the scales example. I could come up with millions more examples.

Such as: Reason can't absolutely prove that the Eiffel Tower or any men definitely exist. I have to assume those things.

But it can tell me that if a man jumps from the top of the Eiffel Tower he will not survive survive uninjured. If your Faith tells you otherwise, you're welcome to try it.

Posted by: Brit at February 28, 2005 12:55 PMBrit:

Prove just "I" and we can move on. No one ever has.

Posted by: oj at February 28, 2005 3:51 PMI think your record's stuck.

Posted by: Brit at February 28, 2005 4:51 PMYes. You can't. So we start with faith. Now you can argue that you like your faith better.

Posted by: oj at February 28, 2005 5:13 PMI'm not hiding from the fact that I can't prove "I". I openly admit and embrace it. Hell, I revel in it.

I've also explained why I don't need to. Now you explain why I do.

Then perhaps we can "move on".

Posted by: Brit at March 1, 2005 5:01 AMBecause everything you say afterwards is a statement of faith, not of Reason.

Posted by: oj at March 1, 2005 7:49 AMReason might be founded on assumptions which cannot be proven, but it's still useful and therefore is a valid approach to problems.

Why isn't it still useful?

Posted by: Brit at March 1, 2005 9:09 AMIt is useful within the constraints you posit.

Posted by: oj at March 1, 2005 9:17 AMAnd within those constraints, and for practical questions, such as calculating which side of the scales will be lower, it is more useful than Faith alone?

Posted by: Brit at March 1, 2005 11:39 AMNo. We don't believe they weigh the same because of reason but because of faith in what our eyes tell us.

Posted by: oj at March 1, 2005 1:47 PMOJ:

I excluded Darwinism because I was inquiring about your viewpoint. You say God told us which. Clearly, they who believe God did so don't agree on what God said.

So, tell me, how do you decide which Faith based explanation is correct?

Brit:

The fact of our existence has no impact upon a materialist. Should we eventually discover we are only brains in a vat, then so be it--another riddle solved.

Ironically, though, it bears very heavily upon the theologically exercised. For if we don't objectively exist as separate individuals, then the whole of Christianity's underpinnings come crashing down.

Reason is therefore superior to faith, because Reason wants to determine what is, rather than merely settle for the thumb-sucking fairy tale du jour.

Posted by: Jeff Guinn at March 1, 2005 4:25 PMOJ:

But we haven't put anything in the scales yet. Our eyes haven't seen anything. We've just predicted, using reason, that the heavier side will be lower.

Posted by: Brit at March 1, 2005 4:36 PMThat's not reason, just experience. That's why children have to be shown it.

Posted by: oj at March 1, 2005 4:41 PMI bet there's lots of children now who've never seen a set of balancing scales. Scales to them are electronic boxes in the bathroom.

I bet the smart ones could work out, using reasoning and simple mathematics, once the principles were explained to them, that the heavier side would be lower.

Posted by: Brit at March 2, 2005 4:10 AMYet they don't. In Kindergarten or first grade someone shows them a scale. We don't reason much, we observe and trust pedagogy.

Posted by: oj at March 2, 2005 7:23 AMThat's probably true, but irrelevant to the point.

The fact that you're arguing about the uncontroversial premises of the argument suggests that you're unwilling to address the controversial conclusion of the argument. Chicken?

Posted by: Brit at March 2, 2005 7:48 AMWhat is the premise?

Posted by: oj at March 2, 2005 8:00 AMThat reason is useful for most questions.

Posted by: Brit at March 2, 2005 8:25 AMYes, just not the significant ones.

Posted by: oj at March 2, 2005 8:29 AMBut it is useful for questions such as:

"Should I jump off the Eiffel tower?"

"Will two identical objects weigh more than one of those objects on its own?"

"How are diamonds formed?"

"Is it valid to say that the Holocaust happened?"

"Which is the best explanation for evolution?"

No, it isn't used for any of those except maybe the diamond one.

Posted by: oj at March 2, 2005 8:59 AMWhy?

Posted by: Brit at March 2, 2005 9:02 AMOrrin gets it right. We experience things.

How does not matter. We do not seem to have any choice in that.

We can then operate on the experiences by reason, or by faith; and that generates further experiences that are predictable to a degree.

Over my computer is a quotation from somebody named Dan Stanford, a wise man whoever he was: 'Experience is what you get when you don't get what you want.'

You can quantify that. Here in America almost everybody, even Orrin, operates on the principle that you get what you want more often by reasoning than by faith. (PZ Myers wrote a hilarious piece about faith-based plumbing)

Pretending otherwise does not change anything.

Posted by: Harry Eagar at March 2, 2005 4:01 PMWe believe we experience them. That's enough.

No one reasons plumbing. They hire some blockhead with experience to handle it.

Posted by: oj at March 2, 2005 4:05 PM