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In mathematics and logic, it seems that once a proof of some theorem is discovered, then it is taken to be "absolute truth" within the axiomatic system from which it was derived. My question is: are we fully justified in asserting this?

For example, suppose we were to assume that the string $$$ is an axiom, and we take as a rule of inference that if a string of dollar signs is a theorem or axiom, then that string with one more dollar sign added on is also a theorem. Then, we might give the following as a proof of $$$$$:

  1. By our rule of inference and our axiom, we know that $$$$ is a theorem.
  2. By our rule of inference and the theorem $$$$, we know that $$$$$ is a theorem.

This proof seems quite trivial, and is incredibly convincing! But, is all we've done convince ourselves? I would bet my life that $$$$ is a theorem, but I still cannot truly say that there are no errors in our proof above. I do not see how we could know that we have properly used our rule of inference, and in particular: how do we know we are justified in using a rule of inference at all? Isn't this a hidden assumption we are making?

It seems to me that there are hidden assumptions in all proofs--there seems to even be an assumption in rigorous proofs that we have not made any hidden assumptions! So, when we "prove" something, are we really just coming up with arguments that convince us, and are just suffering from "logical hubris" when we assert theorems to be true?

I'm by no means a philosopher by training, so please don't assume too much of me! I hope this questions makes sense, thanks.

  • You might like my answer to "What is the difference between Fact and Truth?" – labreuer Jan 12 '14 at 16:44
  • Sounds logic but the whole sentence is an assumption not something that can be considered true or false. – user5231 Jan 12 '14 at 23:46
  • This might be of interest: ams.org/notices/200807/tx080700773p.pdf – Baby Dragon Jan 13 '14 at 1:40
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There are a few things to unpick, here.

First, there's a difference between provability in a formal system, and "truth", which is a question of the relationship between language acts and facts. The statement "Juh mapple Neele" is neither true nor false, but nonsense, unless it is recognised as a poorly pronounced version of the French phrase Je m'appele Niel. The existence of a correspondance between the utterances and facts is necessary to have a meaningful notion of spoken truth. Similarly, your string $$$$$ which is obtainable by transformations from $$$$ is not 'true', except if you provide some correspondance between it and some model. That is to say, you haven't provided any thing for $$$$$ to be true of, nor a notion of what it would mean for it to be true.

Of course, French — and English, and all other natural languages — don't have a formal specification of how they correspond to reality. We more or less learn the relationship between the language and reality by correlations, and by depending on one another to mostly co-operate in reinforcing the conventional correspondance between language and reality. If we tried to describe the correspondance, it would just be with more language; just as we tend to use mathematics to make statements about meta-mathematics, and sometimes try to describe logic by boolean algebras. Our formal use of logic is a carefully honed skill of associations, a refined sort of activity of the same sort as natural language usage is, sharpening that more common skill to scalpel sharpness and rigidity.

Having said this, how do we know that we reason correctly, logically? Lewis Caroll (of Alice in Wonderland fame) wrote precisely on this subject about logical regress in certainty of provability in "What the Tortoise said to Achilles". The moral of the story is: if you are sufficiently skeptical of how to apply logical rules of inference, then any logical conclusion is impossible to reach, because you cannot prove that you have correctly proven something without putting the skepticism about proof simply at one more remove. The rules of inference are best practises for obtaining transformations of sentences in such a way that it minimally increases the error of the statement. In its role in formal derivations, it may be regarded as a sort of dance step — a simple move to be mastered as part of more complicated coreography of precise motions to convey a message, which is interpreted by others who know the correspondance of the formal system to other systems of thinking and imagining, sometimes consisting of physical reality or a caricature of it.

If you are not confident in your ability to make simple motions, e.g. to make simple speech acts, then no linguistic argument will convince you that language has any correspondance to reality; and the same is true of formal logic. What formal logic allows us to do is to check whether an argument has any steps which we don't have very much confidence in. That is, if we are sufficiently skeptical, the best we can do is say that we cannot find obvious flaws in the reasoning; the complementary case being when we can find obvious flaws, as in an invalid derivation. Of course, it is possible to make mistakes when identifying a step of reasoning as invalid, but at least it provides us with the opportunity to try to demonstrate it's falsehood by attempting to imagine a refutation of the statement being asserted by focusing on the consequences of the mis-step. Then we may try to bring our resources to bear, if we are worried that we have reasoned poorly, to ask whether the statement to be proved or the imagined refutation seem more coherent with our models.

Of course, as respects captial-T Truth, one might say that any claim to have access to absolute truth represents an act of hubris: that the world is necessarily so simple as to be completely comprehended by a mind handily contained within a cubic foot of space. To anyone familiar with the phenomena of turbulence and quantum mechanics, or the mathematical construction of fractals or strange attractors, or indeed the Halting problem and the P vs. NP problem — in short, the mathematical discoveries or problems of the 20th century — the idea that absolute capital-T Truth is something which is easily accessible even from premisses of analytical philosophy is laughable, unless the statement is so vague as to communicate almost nothing of the world's complexity, or so elaborate that it exists only as a script which was deliberately and pains-takingly recorded; the knowledge consisting more in the ability to decipher and recite the record as a sort of ritual for the purpose of undertaking some act in the world of which the record somehow represents an incomplete but possibly helpful sketch. Formal logic is a school of such ritual performance, which describes a particularly reliable way to formulate and perform such rituals.

Logic is not a guarantor of truth, but merely (!) a very powerful tool for precise argumentation, and one which has a very good record of reliability, even if many people find it a difficult tool to wield carefully. As with any tool, it is only as reliable as the person who uses it. But somehow, as with the tools of art and of construction, we find that we can often recognise when someone has used it skilfully, and aspire to ideals of not only precise but elegant and artful wielding of this tool — which after all is a form of language.

The role of formal logic is, in short, a form of exercise in which to try to reason carefully, and to learn what careful reasoning looks like. But it cannot teach one to reason, nor guarantee good reasoning, without having some basic skill already.

  • Nice answer. My only suggestion would be to write the paragraph about "What the Tortoise said to Achilles" in big, bold, block letters. It's a shame the story isn't mandatory assigned reading for high-schoolers. – David H Jan 12 '14 at 13:58
  • In my generation, there was a sort of self-organised activity in which it was read by almost everyone who would be able to understand it, as a side-effect of it being included as an early Chapter of Gödel, Escher, Bach. Of course, as with many things that one reads as a teenager, the full ramifications of it only because clear to me much, much later. – Niel de Beaudrap Jan 12 '14 at 14:10
  • Very good, but, unfortunately, you get lost in self-contradictions in the paragraph that starts with: one might say that any claim to have access to absolute truth represents an act of hubris as there is no difference between "absolute true" and just "true" and the above is certainly a claim to have access to a truth (namely, that such an act constitutes hybris). So I recommmend to remove that whole paragraph – Ingo Jan 13 '14 at 0:32
  • @Ingo: Or you could just view that paragraph as an act of hubris, or as making an assertion about the world which is so vague as to communicate almost nothing of the world's complexity. Both, in fact, would be true. – Niel de Beaudrap Jan 13 '14 at 5:07
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    @NieldeBeaudrap See how you can't stop making assertions about truth? :) Seriously, sentences of the general form "Humans can't discover truth." have a bigger epistemological and logical problem than the usual liar paradoxa. – Ingo Jan 13 '14 at 10:56
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I think some Frege/Hilbert ideas might help in making progress on this question, since in many ways, the Frege-Hilbert Correspondences were about this very issue of the nature of mathematical axioms and what proofs using them are really about.

Let's suppose we have a piece of logical syntax constituting an axiom, and a rule of inference that tells you something you can do with this axiom to derive a theorem. Awesome. But what comes next is the question of what kind of mathematical structure this system of inference amounts to. Do we have a Boolean or Heyting Algebra? Do we have a Topos? Do we have a transitive set model?

Here is a model that might completely encapsulate your proposed logical system - the ordered sequence of natural numbers. We have a base zero value ($$$), and a successor operation (adding a $). Since we have this structure already (the numbers 0,1,2,3, ... and so on), there doesn't seem to be any problem whatsoever with saying that your proposed inference scheme is also well founded. Now this may not necessarily be your intended model, but syntactically, there's nothing obvious in your formulation to say there's anything wrong with it as an interpretation. Maybe seeing your axiom/rule framework as a subsystem of Peano Arithmetic might help you find out interesting properties and theorems you might not have considered otherwise. On the other hand, if you want to be more specific about a structure you're interested in, you might consider adding more axioms or rules to home in on the particular behaviour or resources of your model.

Now let's pull it back a bit - if your inference system is fine because it's somehow "not new" relative to our current counting systems, then what is it that makes these pre-existing counting systems or more general mathematical theories okay? The axioms that we take to inform us about number theories tell us something valuable about what it is to count, multiply, factorize etc., and this gives us a practical way of agreeing on answers to outstanding questions of applied practical use in physics, engineering, and even in more abstract ideas of strategic planning and modelling. But why should we agree on them?

Well, I think there're two diverging strands on what it is about our axiomatic systems of mathematics that makes them foundational. The first is to say that in building axiomatic theories and individuating structures exemplifying the axioms we're investigating, what we're doing is describing and inventing a variety of abstract technologies that may (or may not) be useful in whatever contexts we want to apply them in. Mathematics brings a series of tools that might work very broadly and successfully, and that many things in nature and our lives are apt to be reasoned with using the technologies that we invent and apply. This is (a very broad-stroke version of) David Hilbert's perspective, because the mathematician is not specifically aiming at Truth-in-the-world in doing their reasoning, but more just at delving into and unpacking the internal definitions and relations of the different parts of the technologies they're creating.

The second is to focus particularly on certain central theories that have proven to work well, and to develop our understanding of those theories with a view to explaining a core underlying body of mathematical facts. We don't seem to have many different number theories in practical use - we have one notion of what it means to be a number 2, and a simple fact of the matter that 2+2=4. Maybe this tells us something about the general structure of human cognition, or maybe it just tells us something about how the world itself is actually composed of things satisfying a certain structural regularity that we can latch on to; either way, there is some kind of phenomenon that mathematics is trying to be a scientific theory of, and uncovering the nature of that structure is what mathematicians try to achieve in proving theorems in certain accepted fields of mathematical interest. This is more like Gottlob Frege's view, in that mathematical Truth really is out there, and mathematicians are doing their empiricist duty to investigate and test it.

In either case, what's going on is that we're not "just" drawing out cute syntactic relationships. There is some kind of mathematical structuring, relevant to the cognitive power that maths gives us across the other fields of study and practice, that goes into why some inference rules and axioms are accepted and worked with. Understanding logic is about getting a feel for these structures, where they might be usefully employed and their consequences and limitations. Is there an "absolute Truth" notion here? I don't see why either Frege or Hilbert need deny that there is - for Frege, mathematical Truth comes from the role of mathematical abstraction in scientific practice, and for Hilbert, mathematical Truth is about its own systems and formalisms, so both can claim a kind of strict truthfulness in mathematical assertions.

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In science a theorem is a truth until three independant observations do not fit. Until we can make the three observable abberant observations that declare it myth it stands as true.

How is a myth debunked in the land of math? What is your equivalent to a discreet observation in the land of science. The math world world is foreign to me, the culture is not mine I dare not hazard a comment lest I be shooed away as insignificant.

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