The Incompleteness of Mathematics: A Philosophical Inquiry

[robertjford80]

I'm getting interested in mathematics because as a philosopher I am upset with the amazingly poor standards of rigor in my field. I am looking to mathematics for guidance. I would like to formalize philosophy and turn it into a deductive science. I have a deep interest in logic so I decided to study mathematical arguments and figure out what makes them true. Much to my surprise I have found mathematical arguments to be much more informal than I thought they would be. It seems that the rock-bottom foundations of mathematics still have not been made secure. What I'm interested in doing is proving mathematical theorems entirely in a new logical language that I've invented without resorting to notation. Essentially, I'm working on the old Russellian problem of showing that math reduces to logic. I have sort of proven Euclid's first theorem with just words alone but my logical language is still in a state of flux and I keep making adjustments to it. Plus, I have not proven it to my satisfaction. Let's now look at the Euclidean definitions for geometry as translated by Richard Fitzpatrick which amazingly has the original Greek side by side and given that I have a good knowledge of Greek I can sort of work out the original.

A point is that of which there is no part.
Σημεῖόν ἐστιν, οὗ μέρος οὐθέν.

Has any geometer defined part?

And a line is a length without breadth.
Γραμμὴ δὲ μῆκος ἀπλατές.

Has any geometer defined length or breadth?

And the extremities of a line are points.
Γραμμῆς δὲ πέρατα σημεῖα

Has any geometer defined extremity?

You get the idea of what I'm driving at. My theory is that if we understand the meaning of these words then we can make deductions to prove Euclid's theorems without even saying an image of what he's referring to.

 

[hilbert2]

Modern mathematics studies logical structures, which have a certain set of rules called axioms, from which theorems can be deduced logically. For example, the German mathematician David Hilbert created (in late 19th century) a set of axioms for geometry, that made a fully rigorous approach to geometry possible. Mathematics was not properly axiomatized before 19th century and definitely not in Euclid's time.

Euclid attempts to define what "points" and "lines" are, while in modern mathematics the objects studied are completely defined by the rules they obey, without them having to correspond to anything from the real world. If we renamed "points" and called them "apples" and renamed the "lines" to "oranges", the axioms would still describe exactly the same logical structure.

Questions like "would numbers exist if no one had ever invented them?" are an example of something that is philosophy, not mathematics. An example of a mathematical question is "do the axioms of geometry contain redundant axioms that would be possible to deduce from the other axioms?"

 

[SteamKing]

It has been demonstrated, by Kurt Godel and others, that formal systems like arithmetic and geometry contain some basic elements, called axioms (or postulates in geometry), which, while assumed to be true, cannot be proven within that formal system. Godel's Incompleteness Theorem shook up mathematics as much as philosophy when it was announced in 1931:

http://en.wikipedia.org/wiki/Kurt_Gödel

 

[KenJackson]

I'm always pleased to see the invocation of Kurt Gödel's Incompleteness Theorem.

I don't understand the math, but philosophically it seems like a needed boundary. Many would love to completely prove everything with extreme rigor right down to what a point is. But Incompleteness says Stop! You can go this far and no further. It's analogy to Heisenberg's Uncertainty Principle seems apt. Those two tell us where the walls of our existence lie (or at least of discoverable knowledge). I find that comforting in this universe with no edge.

 

[SteamKing]

As an aside to the OP, the Loeb Classical Library, which is published by the Harvard University Press, publishes pretty much all of the classic books of ancient Latin and Greek literature, and each volume contains the original Latin or Greek text with a parallel English translation. In terms of mathematics, you will find Euclid's Elements along with the works of Archimedes and others from the ancient world.

http://www.hup.harvard.edu/collection.php?cpk=1031

As a philosopher, I'm sure that the OP might be interested to study the works of the Roman and Greek philosophers in the original.

 

[AlephZero]

I'm sure the OP has heard the comment that "the entire history of western philosophy is just footnotes to the works of Plato", but math has made a bit more progress in the last few thousand years.

If you want an easy-to-read update on axioms for geometry, try something like Hilbert's "foundations of geometry" https://archive.org/details/abr1237.0180.001.umich.edu

Or just jump in at the deep end, with http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory

 

[robertjford80]

Mathematics was not properly axiomatized before 19th century

I would argue that it is still not properly axiomatized, rather it was just much more axiomatized in the 19th century than in the 18th and it still needs more axioms and definitions. In any case, who were the mathematicians that made this happen? Are you talking about Peano's axioms, as well as Dedekind's/Cantor's work?

As far as Gödel goes, I believe Gödel's theorems are false. I have a slipshod proof of this but I admit that I do not have the necessary knowledge of this yet to be fully confident of my proof. It is good however to sit down and think up a proof so as to orient yourself better when you're reading things.

 

[AlephZero]

I believe Gödel's theorems are false.

Hmm... it was another philosopher (Russell) who defined "belief" as "that for which there is no evidence".

 

[SteamKing]

I would argue that it is still not properly axiomatized, rather it was just much more axiomatized in the 19th century than in the 18th and it still needs more axioms and definitions. In any case, who were the mathematicians that made this happen? Are you talking about Peano's axioms, as well as Dedekind's/Cantor's work?

Augustin-Louis Cauchy was the first modern mathematician to 'stress the importance of rigor in analysis.' In particular, the calculus had been used for some 150 years before Cauchy began to put it on a logical foundation.

http://en.wikipedia.org/wiki/Augustin-Louis_Cauchy

As far as Gödel goes, I believe Gödel's theorems are false. I have a slipshod proof of this but I admit that I do not have the necessary knowledge of this yet to be fully confident of my proof. It is good however to sit down and think up a proof so as to orient yourself better when you're reading things.

IMO, you are sailing into a headwind on this contention. More than 'slipshod proof' will be required to overturn Godel. It is important to note that we are not talking about "Godel's Hypotheses" or "Godel's Conjectures", but "Godel's Incompleteness Theorems". Godel has published proofs of these theorems, which have been examined and accepted by other mathematicians. In order to invalidate these theorems, you must find and establish some logical error contained therein, which is a mighty tall order, IMO.

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

 

[AlephZero]

IMO, you are sailing into a headwind on this contention. More than 'slipshod proof' will be required to overturn Godel.

A few philosophers have been there already, e.g. J R Lucas, and Roger Penrose.

But there seems to be a basic flaw in all their arguments, namely that they can't tell the difference between a mathematical theorem and a human being.
http://www.iep.utm.edu/lp-argue/

 

[pwsnafu]

It has been demonstrated, by Kurt Godel and others, that formal systems like arithmetic and geometry contain some basic elements, called axioms (or postulates in geometry), which, while assumed to be true, cannot be proven within that formal system. Godel's Incompleteness Theorem shook up mathematics as much as philosophy when it was announced in 1931:

http://en.wikipedia.org/wiki/Kurt_Gödel

No. Godel's theorems have nothing to do with axioms, as axioms all have a trivial proof (i.e. verifiable) in that formal theorem. Godel's incompleteness refers to statements whose truth are undecidable. That is you don't know if they they are true or false.

I'm starting to think we need a FAQ on this topic.

More to the point, Euclidean geometry is not susceptible to the incompleteness theorems because it does not contain PA. Specifically, Tarski's axioms of Euclidean geometry have been proven to be consistent and complete.

 

[SteamKing]

No. Godel's theorems have nothing to do with axioms, as axioms all have a trivial proof (i.e. verifiable) in that formal theorem. Godel's incompleteness refers to statements whose truth are undecidable. That is you don't know if they they are true or false.

I'm starting to think we need a FAQ on this topic.

I beg to disagree.

http://en.wikipedia.org/wiki/Axiom

"An axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Within the system they define, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow otherwise they would be classified as theorems."

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

"Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem.

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency."

 

[pwsnafu]

From this thread

To put the above answers more explicitly: A proof of a sentence S in an axiom system means that (informally):
(a) there is some relation which can be interpreted as implication in the system
(b) there is some rule(s) (e.g., Modus Ponens) allowing the axioms to be combined to give other (not necessarily new) sentences.
(c) from the axioms and rules one can create a chain of implication that starts from a subset of the axioms and ends with your sentence S.
So, if you have a set of axioms, as long as "A implies A" is valid for all A (as it should be for implication), then your axioms are automatically proven.

Of course, it is possible to prove the axioms from other propositions in your theory (the set of consequences of the rules applied to the axioms), but this becomes circular.
If you want prove the axioms from some other basis besides those propositions in your theory, then you are running contrary to the definition of axioms.

Also "true but unprovable" is a case of getting your definition of completeness mixed up.

 

[jgens]

SteamKing: There is a lot of terminology here and I will do my best to clear it up.

1. There are numerous ways of defining "axioms" and it would benefit to put these notions in the correct context. The incompleteness theorems deal with formal theories so this is the proper place to start. Supposing we have fixed a formal language a theory T is just a set of sentences in this language (called axioms) while the theorems of T are just sentences such that T⊦φ. In this sense, the axioms of T are theorems and they are provable!
2. The claims regarding the incompleteness theorems in that Wikipedia article are a bit sensationalist in my opinion and certainly deserve some clarification. Having recently done some reading on the issue, the "true sentences" in that claim concern the true sentences in the standard model of arithmetic. So essentially the claim at hand here is the standard model of arithmetic contains theorems which are not theorems of PA. But since we can find (non-standard) models of arithmetic not satisfying these undecidable theorems in PA this usage of "true" is quite misleading. Regardless the incompleteness theorems say considerably more than this and the proper emphasis should be on undecidability.

Hopefully this clarifies the misunderstanding.

Edit: For further clarification

Also "true but unprovable" is a case of getting your definition of completeness mixed up.

What I thought was meant by "true statements" in that thread is not quite right unfortunately What people have been meaning is true sentences in the standard model of arithmetic instead. This technically saves their claim, since there are true sentences in the standard model of arithmetic that are unprovable from PA, but makes it rather sensationalist in turn. In particular, model theoretic considerations allow us to find non-standard models of arithmetic not satisfying these sentences, so these sentences are certainly not true (but rather undecidable) in some models. So the furor really becomes "there are sentences in models of PA which are not provable in PA" which is really not saying very much, since the phenomenon where models of a theory prove strictly more than the theory itself is fairly common.

 

[SteamKing]

All this may very well be true; I'm no logician nor have I studied any of this in detail.

However, the OP way back was asking if anyone had ever defined terms like 'length' and 'breadth' and such as they are used in developing geometric ideas. All I pointed out was that arithmetic and geometry are based on these things called 'axioms' or 'postulates' which are assumed to be true. Certainly, most of Euclid's postulates have been accepted as true with the exception of the famous Fifth, or Parallel, Postulate. Basic geometric objects, like points, lines, planes, etc., have been described with certain properties like 'length' and 'breadth' which don't have much in the way of a formal description, except what we understand from the crude physical analogs which we use to represent them.

I merely pointed out to the OP my understanding that geometry and arithmetic, by themselves and consistent with what is known about their characteristics, properties, etc., cannot be completely developed without reliance on these assumed axioms. Godel's work on Incompleteness followed from the famous program posed by David Hilbert in 1920. The upshot, as I understand it, was that the basic axioms used to develop arithmetic and geometry could not be proven entirely within the logical structure of the subject itself.

http://en.wikipedia.org/wiki/Hilbert's_program

 

[robertjford80]

IMO, you are sailing into a headwind on this contention.

Yes, I'm aware of this which is why I'm not going to bother trying change anyone's minds now. Once, I've got more sophisticated tools I'll devote myself to the task but not now.

In any case, it is true that there must be one fallacy that is the most widely believed and I think Gödel's theorems have that title.

 

[robertjford80]

Tarski's axioms of Euclidean geometry have been proven to be consistent and complete.

This was very helpful. I'll have to look into this.

 

[pwsnafu]

Yes, I'm aware of this which is why I'm not going to bother trying change anyone's minds now. Once, I've got more sophisticated tools I'll devote myself to the task but not now.

In any case, it is true that there must be one fallacy that is the most widely believed and I think Gödel's theorems have that title.

I'm not sure what you are trying to say. Because we have explicit examples of unprovable statements of Peano arithmetic. Goodstein's theorem is probably the best example.

 

[D H]

I merely pointed out to the OP my understanding that geometry and arithmetic, by themselves and consistent with what is known about their characteristics, properties, etc., cannot be completely developed without reliance on these assumed axioms. Godel's work on Incompleteness followed from the famous program posed by David Hilbert in 1920. The upshot, as I understand it, was that the basic axioms used to develop arithmetic and geometry could not be proven entirely within the logical structure of the subject itself.

You still misunderstand. It's not about the axioms per se. The axioms of a mathematical system are what they are. Gödel's theorems were on the consequence of those axioms, to wit: Is a mathematical system complete and consistent?

A mathematical system is complete if every statement that can be made in the context of that system can be proven or disproven solely via the axioms of that system. A mathematical system is consistent if it admits no contradictions. The answer to the above question is "it depends". Some systems can be proven to be both complete and consistent, but others cannot. In fact, certain mathematical systems must be either incomplete or inconsistent.

Hilbert's ultimate goal was not to prove the axioms of mathematics. It was to reduce mathematics to logic and to prove that mathematics is both complete and consistent. There's not one word in Hilbert's program or in Gödel's theorems about proving the axioms. That doesn't make sense. The axioms of a mathematical system are the statements one assumes to be true. Proving an axiom is trivial because the axiom is a given.

I'm not sure what you are trying to say. Because we have explicit examples of unprovable statements of Peano arithmetic. Goodstein's theorem is probably the best example.

Another example of unprovable statements is the continuum hypothesis. Proving or disproving the continuum hypothesis was Hilbert's first problem. It turns out this cannot be done in the context of ZFC. Until Cohen's work, Gödel's theorems were viewed as perhaps a peripheral concern. That Hilbert's very first problem was a shining example of incompleteness is what made people pay very serious attention to Gödel's theorems.