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On Kurt Gödel's essays, “Some Basic Theorems on the Foundations of Mathematics and their Implications” (1951) and “The Modern Development of the Foundations of Mathematics in Light of Philosophy” (1961).
Gödel is famous for some "incompleteness theorems" of direct interest only to those trying to axiomatize mathematics. What are the implications for the rest of philosophy? Gödel thought that, contra people's claims that incompleteness demonstrates some kind of relativism, he had showed that the world of mathematics even in its most abstract reaches is part of the real (though non-physical) world. Mark, Wes, Dylan, and guest Adi Habbu try to figure out these unpublished and dare I say incomplete essays.
Listen to Adi's introduction, read more about the topic, and get the essays.
End song: "Axiomatic" from Mark Lint & the Simulacra. Read about it.
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The predictable absence of mention of the prominent phil/math thinker of the last century, Whitehead, screams out for inclusion in such philosophical activity as presented in the episode. How can Russel’s mathematics professor, his mentor, and THE lead worker on _Principia Mathematica_ be ever-ignored by those who – I assume now – claim to be unbiased in their philosophical activities.
What Dylan discusses around 45 min. is integral to the argument Whitehead had w/ Einstein about blindly mistaking your math stick-model of the physical universe for what is actually happening: and so the establishment cadre of Einstein’s physicist fan-boys ridiculously adhere to the ‘given-ness’ that matter curves empty space.
Einstein is to physics what Russel is in philosophy – unquestioned heroes of the two academic fields. Someone needs to study the impact this has on academic honesty and even research funding.
Whitehead dug into math and natural science with rigor unmatched by his peers -evidenced by Harvard recruiting him to come and lead its philosophy program following in the legacy of no less than W. James. ANW’s ubiquitous statements on misplaced concreteness and excessive abstraction (which are simple language equivalent to what Godel said), his nod to a sort of neo-platonism, his tutelage of so many modern philosophers and artists, and the unsettling impact his mantra of ’embrace uncertainty’ evidently had on students like Rorty and Quine – not to mention Quine’s capitulation to the inadequacy of the Bertrandesque demand for nothing but analytic philosophy.
With such a 20th century academic powerhouse deeply active in all the major philosophical, mathematical, and scientific work of his (and still our) times, why does Whitehead continue to be ignored???
Gödel once wrote to an editor:
‘It is easy to allege very weighty and striking arguments in favor of my views, but a complete
elucidation of the situation turned out to be more difficult than I had anticipated.’
In my opinion the arguments presented by Gödel in favor of platonism does not actually work, but i still think the questions about the status of mathematical reality are really interesting and it is somewhat a shame they got tied up in technicalities, when they should have been examined more thoroughly. For after all what does it mean to say that it is true that 1 +1 = 2? And what does this fact hinge on?
The best counterargument to the platonistic or realistic view is simply that it can’t hinge on some kind of mathematical reality since there exists no such actual thing.
On the other hand the realist persuading one to accept the realist view can simply say “it works”, for after all who needs a formal consistency proof for say Peano arithmetic when you can prove it like this:
1. Obviously all the propositions contained in the axioms are true (given their intended interpretation)
2. The mehod deriving theorems from the axioms are truth-preserving
3. A contradiction is always false
Since you cannot derive a false proposition you cannot derive a contradiction
Since you cannot derive a contradiction the system is consistent
But then again what does it mean to say that it’s true that “For every natural number n, S(n) is a natural number”? Is it just an arbitrary definition like “Emerald city buildings contain green tiles”? There seems to be some kind of difference but what kind?
Of these things Gödels theorems alas tell us nothing, and they are best avoided unless the waters be muddled.
K, you’re absolutely right that this is only one attempt at approaching the concept of mathematical Platonism, and there’s much more to be said about what metaphysics would be involved in affirming the “reality” of numbers. Do you (or anyone else) have a good recommendation for a reading (preferably by an author that most of us will have heard of, but not necessarily) on this topic? I don’t know when we will next get around for more philosophy of math, but I think that topic is a good one for a 3rd go-round in this area, and very in line with the more systematic discussion of metaphysics I’d like to see come out in more of our future episodes.
I must recommend Torkel Franzéns PhD thesis “Provability and Truth”, it’s available at his archived web page among other stuff:
http://web.archive.org/web/20070830005655/http://www.sm.luth.se/~torkel/index.html
It’s an interesting introduction or summary to the philosophy of mathematics, easy to read and short, but well thought out and argued. It’s like a mountain well clear and deep.
Here’s what Solomon Feferman wrote about it:
“I find the ideas expounded in his thesis that he calls “classical
eclecticism” very thought-provoking and deserving of wider
consideration.”
I’m not a philosopher, but my view was that it was psychological. It’s a function of the fundamental way we construct meaning and reality.
You’d know what I was talking about if only you had made love to the sountrack of the proof of Fermat’s last theorem!
http://www.egs.edu/faculty/alain-badiou/articles/infinity-and-set-theory/
Great podcast but I think you could have been a bit more careful to distinguish between arithmetic and set theory. Arithmetic truth, though not computable, is at least definable in the language of set theory. Arithmetic Platonism is therefore a coherent position. On the other hand what does it mean to be a set-theoretic Platonist? This is far more problematic since ZF can not construct a model of itself (by the second incompleteness theorem). Some set theorists do defend a set-theoretic Platonism, but this position is far more controversial.
I’m no expert on this topic so bear with me.
Couldn’t you just extend ZF producing a theory ZFT defining a truth predicate for ZF?
Then would set-theoretic Platonism be a coherent position? Since set-theoretic truth would be definable in ZFT?
Unfortunately it doesn’t make sense to talk about a truth predicate for ZF because ZF is a (incomplete) theory, a set of sentences in first order logic. We talk about truth with respect to models. The existence of a standard model of the natural numbers with addition and multiplication is what allows us to talk about arithmetic truth. Set theory, by contrast, has no such standard model. It has instead a rich multiverse of non elementarily equivalent models, none of which can claim to be the “standard” or true interpretation of ZF. Here’s a good lecture on the set theory multiverse https://www.youtube.com/watch?v=WndanxPlDFk
Hey, Nick, it might behoove us to do an episode specifically on set theory in the future. Send me an email and we’ll chat: mark@partiallyexaminedlife.com. -Mark
Hmm I see.
I wonder if there is there’s two problems for platonists here:
1. There exists no standard model of a theory T
2. It is unknown if there exists any model at all for theory T
(btw would it be correct to say that the existance of a model of T is equivalent to con(T)?)
I can see why it would be strange to talk about truth for theory T (or a model of T to be more specific) if 2 were true, but if only 1 were true i don’t see the problem for platonists? Isn’t the fact that there exists a standard interpretation of arithmetic just historical or psychological? Also couldn’t the platonist say that by true they simply meant true in all models of T? Then there would be no reason to care what the standard model where?
I will watch the lecture, you should write a blog post about set-theory and the foundations of mathematics as a follow up on this episode nick so that i can ask even more stupified questions.
I think that’s exactly the right distinction to make. I probably muddied the water a bit by saying that “ZF cannot construct a model of itself”, which, while true, might be irrelevant for the Platonism/formalism debate since models can be constructed meta-theoretically. There are some very sticky theory/meta-theory distinctions that are confusing to non-experts like me.
In answer to your parenthetical question: Yes, T is consistent if and only if there exists a model for T. Since set theorists talk about models of ZF, you might wonder what prevents them from constructing these models inside ZF and thus violating the second incompleteness theorem. Here’s a stack exchange post about that: http://math.stackexchange.com/questions/772116/why-doesnt-v-is-a-model-of-zf-imply-consistency-of-zf
Actually both 1 and 2 prevent us from talking about absolute set-theoretic truth. This is because the different models of set theory disagree on the truth of certain propositions (the continuum hypothesis, for instance). This is what I meant by the phrase “non elementarily equivalent,” which I should have explained earlier.
That said, there might be two responses from the Platonist. First, she might say that “pluralism” is not incompatible with Platonism any more than a cosmological multiverse is incompatible with the position that material objects exist in the physical universe. Second, (and this is particularly amazing to non-logicians like me) it seems that some set theorists somehow come to have intuitions about the truth or falsehood of ZF-independent statements like the continuum hypothesis. Somehow they have such deep intuitions about the concept of set that they’re able to reason beyond the axioms, something which may seem a bit like divination to other mathematicians. Perhaps in a hundred years experts will accept “not CH” with a private uneasiness similar to that felt by today’s mathematicians when they invoke the axiom of choice. Here’s an interesting summary: http://en.wikipedia.org/wiki/Continuum_hypothesis#Arguments_for_and_against_CH
(The comparison of CH to the axiom of choice might also suggest a third response from the “instrumentalist” position, since the truth of the axiom of choice was essentially settled by its usefulness to the rest of mathematics.)
“This is because the different models of set theory disagree on the truth of certain propositions”
So does different models of PA, it’s just that when you say arithmetic truth you mean true in a specific model N of PA.
What is to stop the plationist from saying: well by set-theoretic truth I mean true in model T of set theory?
The only difference I can think of in the two cases is that arithmetic truth already has a agreed on meaning outside of set theory, so that when choosing models of PA that describes true arithmetical statements the one choosen will be the one in accordance with the already agreed upon meaning of arithmetical truth.
On the other hand since there is no general agreed upon meaning for what set-theory is, there is no standard model of set-theory.
This also explains why: “Somehow they have such deep intuitions about the concept of set that they’re able to reason beyond the axioms, something which may seem a bit like divination to other mathematicians” it’s not that they reason beyond the axioms, it’s just that the axioms are an incomplete description of the thing they are thinking about, therefore they can draw conclusions about the thing that do not follow from the axioms.
Could math not bust be a modelling tool, a story that we tell that allows us to paint a useful picture of the world? This could square the circle of empirical observation and human creation, where the model incorporates experience and can be, in principle. a completely human creation.
By the way, doesn’t Wittgensteiln reduce all analytic a priori statements to tautologies, and as such maths is tautological. Why does this tautological nature matter? It would be like invalidating logic?
Math is not ‘analytic a priori’, but synthetic a priori. One might disagree, but the usage of these terms has always been wrapped up with categorizing math as synthetic a priori, ever since Kant started the discussion…and in any case, it’s very easy to arrive at the point that math is not ‘tautologies’. Tautologies are logically true, in the technical sense, and math is not, — math is, as they say, ‘logically satisfiable’. The notion of satisfiability needs, I assume, no introduction…?
Probably that doesn’t clear things up, as far as your ‘btw’ reference to Wittgentstein, but at least I’ll add that I think Wittgenstein gets into some very interesting stuff about logic.
The question whether there is some distinction between math and logic has multiple answers (there’s not some up-to-date 21st century consensus here; in fact foundations of mathematics is a royal mess).
How could you guys not mention my video on the very lecture you discuss?
https://www.youtube.com/watch?v=cG7MyZtGSB0
There is a no music version avail too…check out all of my Godel videos…I talk about a lot of the same stuff you guys do. I have been asking for a Godel episode for years…glad you finally did one. It was a good introduction to Godel…bring me on next time I could have cleared up a lot of your confusions…thanks!
Another interesting philosophical romp. While Godel trumps Kant mathematically, Kant clearly trumps Godel metaphysically. While Godel provided a fascinating grasp of the possible foundations of mathematics, he prioritized the mathematical over the metaphysical, the conceptual over the ontological, the Platonic over the Aristotelian, and did not learn from Kant’s concept of intuition.
His confusion led to the inverted implication that somehow there is an intuition or experience of concepts themselves. I certainly remember late night efforts to resolve problems in geometry in high school, sometimes waking in the morning excitedly with the solution, but that is no reason to confuse experience with concept.
While Godel claims to have interpreted Kant most accurately, his interpretation is not only incomplete, but uncomprehending.
Thanks guys for another great intellectual adventure.
Does this quote address the ontological misunderstanding and over-hyped implications of the Liar’s Paradox and misapplication of Godel’s ideas on systems??
“But in the prevalent discussion of classes, there are illegitimate transitions to the notions of a ‘nexus’ and of a ‘proposition’. The appeal to a class to perform the services of a proper entity is exactly analogous to an appeal to an imaginary terrier to kill a real rat.”
I am fascinated by Godel and the philosophy of math, and I am still getting a lot out of this podcast episode. Just for the record, I am a non-mathematician, an intuitive visual thinker, who is nonetheless drawn into these higher and higher levels of abstraction and fascinated by the math that was not taught well to me in school..
In my view,the rough, non-technical description of Godel’s theorems would be that “you have to step outside the frame to get the whole picture.” But I am also aware that I am a philosophical puppy nipping at the heels of giants, so…. carry on!
Thank you guys, the discussion was very good and insightful. The guest speaker also did an excellent job, kudos. Very good episode. You approached a very technical philosophical context and topic with due sagacity, I appreciate that.
Mark and the others chuckle (rightfully) at some of the lazy post-modern/relativistic extensions of incompleteness to domains outside of axiomatic mathematics. I don’t, however, believe you discussed the possible connections of Godel’s incompleteness to Turing’s Halting problem. Isn’t a weakened version of Incompleteness (check out http://www.scottaaronson.com/blog/?p=710, or even wikipedia en.wikipedia.org/wiki/Halting_problem#Relationship_with_G.C3.B6del.27s_incompleteness_theorems) a consequence of the undecidability of the halting problem?
If you buy this, take it a step further. Many NON-postmodern, analytical philosophers subscribe to Informational or Computational ontologies of the world (I don’t). I’m sure someone could take a computational ontology, strap it to the Halting problem somehow, and come to see incompleteness as a fundamental aspect of the world at large, beyond axiomatic mathematics.
Anyway. I love you guys. Anyone who calls a guest of David Chalmers’ reputation insane has my vote.
Thank you guys for another incredibly enjoyable discussion on a topic I thought that nobody could say anything interesting about. I did find out a lot from the episode, but I also feel like you missed the main problem.
You can look at the Godel statement not as self-referential or contradictory, but as a mathematical statement about a number. You can check any number you think of to see if it “works”, but you can’t prove there is a number that works or that there is no such number. Obviously if we stumble across a number that “works” then we have proved there is such a number, so it seems like we know deep down that there is no such number. You can follow Hofstadter and say that we don’t really know because “number” is not well-defined, or you can follow Penrose and say that we know because we have access to special knowledge. Neither option looks good, but I think you do need to pick one or the other.
The existence of a number x satisfying x*x = -1 is also undecidable in a particular formal system. In the sense of Penrose, we can know that this assertion is true based upon our intuition about real numbers, because the product of any real number with itself is never negative. Similarly, we can know that Godel’s proposition G is true, because any natural number satisfying P(n) corresponds to a proof of not G.
But if we can know that x*x = -1 has no solution, then we know that imaginary numbers do not exist, so all of the time we spend formulating the theory of complex numbers is wasted, and the formulation of Quantum Mechanics in terms of Schrodinger’s equation is not available. We can formulate QM without imaginary numbers; otherwise, we couldn’t simulate QM systems on a digital computer; however, the formulation is less elegant.
For that matter, we can formulate physical systems and systems more generally without irrational numbers, even with only natural numbers, even with only a finite subset of natural numbers, and we essentially do so when we program digital computers to simulate systems incorporating the numbers we imagine as “real’.
Godel’s undecidable proposition G is “true” only in a metamathematical sense. More precisely, another proposition within a metamathematical system, corresponding to G within PM, is true within the metamathematical system. As a statement about numbers within the formal system PM, G is not true at all, so it is not “true but unprovable” either. “True but unprovable” is an unfortunate summary of the result. G is undecidable within PM but can be decided either affirmatively or negatively by extending PM.
Deciding Godel’s proposition affirmatively appeals to the intuition in the way that denying the existence of imaginary numbers appeals to the intuition.*
Deciding the proposition negatively does not contradict the metamathematical assertion, because the metamathematical assertion is true of the proposition within PM, not of the proposition within the extended system.
Godel essentially numbers proofs of statements about numbers, i.e. he defines a natural number corresponding to each proof of a statement about natural numbers within a particular formal system, PM. Then he constructs a predicate with one free variable, P(n). P(n) is either true or false depending upon the number n. P(n) is constructed so that if P(n) is true, then n corresponds to a proof of the proposition G, not (there exists n satisfying P(n)). The negation of G is “there exists n satisfying P(n)”.
If G is provable, then no number satisfying P(n) exists, but if this number does not exist, then no proof of G exists; therefore, G can have no proof if PM is consistent.
If not G is provable, then a number exists, and this number corresponds to a proof of G; therefore, not G can have no proof if PM is consistent.
If PM is consistent, neither G nor not G is provable within PM, so PM is either incomplete or inconsistent. G is undecidable, or PM is inconsistent. This summary of the result is more accurate than “G is true but unprovable.”
G is often summarized with “I am not provable,” evoking the liar’s paradox. A proof of G contradicts G, and a proof of not G is a proof that G is provable, which also contradicts G; however, “I am not provable” is not a proposition within PM. Within PM, G is only an assertion about the existence of a number, and the sort of “number” it references is the sort of number Godel uses to enumerate proofs within PM. Asserting the truth of G asserts that none of these “natural” numbers corresponds to a proof of G, which is generally true.
Asserting the falsehood of G asserts another sort of number. In Godel, Escher, Bach, Hofstaedter calls these numbers “supernatural”. Assuming supernatural numbers and the falsehood of G does not contradict G, because a supernatural number satisfying P(n) does not correspond to a proof within PM.
*Deciding “not there exists x such that x*x = -1” true appeals to the intuition. Denying the existence of imaginary numbers appeals to an intuition about numbers corresponding to numbers along a line with positive numbers to the right of zero and negative numbers to the left. Given the rules of multiplication we learn, none of these numbers could satisfy x*x = -1; however, we can extend the idea of “number” to incorporate entities satisfying predicates of this sort, and we can visualize these “imaginary numbers” along lines orthogonal to the “real number line”.