Yesterday I was talking to one of my mathematics professor regarding the notion of proof in general (whatever the word "general" means to the reader).

In short my claim was,

We can only be certain at the cost of certainty.

Now, let me explain how I came to this seemingly paradoxical idea. Please correct me if my argument is wrong.

When we are talking about the conception of proof, we must be (at least it seems to me) talking about the conception of proof in a language. It seems to me that a proof may be defined (loosely speaking) as a certain set of sentences. However, the problem lies in this linguistic conception of proof. Suppose that in a language L we have proved something. Say, for example, we have proved a mathematical theorem in a formal language. How can we say that the proof is correct? Well, we can say that because we have followed only the logical rules to arrive at the conclusion from premise(s). But the question is, how can we be sure that we have followed "only the logical rules to arrive at the conclusion from premise(s)"? More precisely, how can we be sure that we can indeed use the word "only"? If we can't then we need to prove that we have actually proved what we claim to have proved. The process can be repeated infinitely many times.

When I argued like this the professor told me that he thinks that the question is merely wordplay. When I asked him the reason he wasn't able to give me an answer that satisfies me.

So, basically my questions are two,

  1. Is my question only wordplay? If so, why?

  2. If not then am I correct in claiming what I have claimed at the outset?

  • As worded, this is a variant on is my claim correct... but I do think you might be heading towards a very legitimate question. – virmaior Jun 8 '16 at 6:52
  • See @jeffy's answer. If Carrol's concerns on this problem were good enough for Mind magazine it ought to be good enough for your professor. – nir Jun 9 '16 at 6:01

You seem to have discovered the "regress argument". Here is a good explication of the argument and some answers to it: http://www.csus.edu/indiv/g/gaskilld/intro/epistemology3.htm

Your professor was too dismissive of your concerns; this is not just wordplay, it is a known philosophical paradox.

ETA: You may want to see also Lewis Carroll's more entertaining version of the argument: https://en.wikipedia.org/wiki/What_the_Tortoise_Said_to_Achilles

  • +1 on Carroll. I wanted to answer with Carrol's Tortoise and you beat me to it :) I think he should show that paper to his professor. If it was good enough for Mind magazine it is good enough a concern for anyone including a serious professor. – nir Jun 9 '16 at 5:59
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    @nir Thanks. Yeah, a pet peeve of mine is Ph.D.s (presuming this math professor is) who don't understand what the Ph. part is supposed to mean. :-( – Jeff Y Jun 9 '16 at 14:19
  • I think that even though a proof may be well-defined in a formal system. We can't say for sure that something is actually proved within the system. For that we need some "external agency" (for example, in case of computers, it my be us who are checking whether we have followed the correct syntax for programs). – user 170039 Jun 9 '16 at 17:12
  • So even though the conception of a proof may be well-defined within a formal system, we can't say for sure that any proof in that system is actually a proof unless it is "checked" by some "external agency". The external agency convinces "himself" that the proof is correct by devising some reasonable argument which can actually convince himself (and which we may call a "proof" to "him"). – user 170039 Jun 9 '16 at 17:13
  • @user170039 I'll repeat something I said in the chat on another answer: I think you are hitting upon what Hofstadter called "JOOTS" in "Gödel, Escher, Bach" -- jumping out of the system. As in "who proves the provers?" (nach "who watches the watchers?"). You may want to consult that book. I believe Hofstadter is still trying to do such type of work using computers in his academic job today. – Jeff Y Jun 9 '16 at 20:45

A proof in mathematics as well as an "argument" in general has also a "social" aspect.

See Yuri Manin, A Course in Mathematical Logic for Mathematicians (2010), page 45 :

A proof becomes a proof only after the social act of “accepting it as a proof.” This is as true for mathematics as it is for physics, linguistics, or biology. The evolution of commonly accepted criteria for an argument’s being a proof is an almost untouched theme in the history of science. In any case, the ideal for what constitutes a mathematical demonstration of a “nonobvious truth” has remained unchanged since the time of Euclid: we must arrive at such a truth from “obvious” hypotheses, or assertions that have already been proved, by means of a series of explicitly described, “obviously valid” elementary deductions.

Thus, the method of deduction is a method of mathematics par excellence.

[...] Every proof that is written must be approved and accepted by other mathematicians, sometimes by several generations of mathematicians. In the meantime, both the result and the proof itself are liable to be refined and improved.

The historical "stability" of the criteria for an "acceptable" proof does not imply that mathematics and proofs are super-human: they are human (and social) activities.

Thus, regarding:

Say, for example, we have proved a mathematical theorem in a formal language. How can we say that the proof is correct? Well, we can say that because we have followed only the logical rules to arrive at the conclusion from premise(s).

we have to add: and the process we have followed in proving it has been "reviewed" and "agreed" by the "community" of mathematicians and logicians.

  • So, fundamentally proofs are only "relatively true" (say, relative to the language and rules of deduction among others)? – user 170039 Jun 8 '16 at 7:35
  • @user170039 - yes, but... this is the only "truth" we have, and most math truths - as per Manin's point of view - are "here with us" from millenia; so they are still the best "approximation" available to the ideal of truth. – Mauro ALLEGRANZA Jun 8 '16 at 7:49
  • So is my claim correct? – user 170039 Jun 8 '16 at 7:54
  • @user170039 - if you mean "absolute", i.e. "unconstrained" certainty: YES, I do not believe that we can attain absolute certainty. But this does not mean that we have an infinite regress: we stop it when we reach axioms and logical rules. – Mauro ALLEGRANZA Jun 8 '16 at 8:36

The other posts talk about the regress argument, but I think your problem is of a different sort.

The physicist, for example, has to go through a process of creating a mathematical model of whatever "real world" objects he studies, and ponder how well the model reflects the real object.

You seem to have the analogous worry about the process of creating a mathematical model of an instance of "real world reasoning", and how much the model really tells us about the reasoning.

I believe this to be a valid concern that I don't think I've seen anything written on (not that I've looked). But I think it's in the category of things that are widely believed to behave as they seem so there's not much to worry about. Also it's sort of in the reverse from the physicist's problem, since the "real world reasoning" is being used to describe a mathematical object.

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    There is a lot written about this. (As a very accessible touchpoint, I will also point you at math.berkeley.edu/~kpmann/Lakatos.pdf.) The whole enterprise of investigating formalizations that starts from Boole and Frege and leads through Goedel and into Wittgenstein is basically about this. And it is just an obsessively rushed climax to a long story going back at least to Pyrrho. – jobermark Jun 8 '16 at 17:42
  • Or in other words, "What validates the model we're using of the human reasoning process?" ? And ultimately then, "What is 'validation' anyway?" I'm not seeing a real distinction of this from the "regress argument", but it's all interesting (and not "just wordplay" in any case)! – Jeff Y Jun 9 '16 at 15:00
  • One "division point" where the "standard model" sometimes (often?) gets challenged is that old "law of the excluded middle". youtube.com/watch?v=mscjvqtBWIQ – Jeff Y Jun 9 '16 at 15:09

If the argument is truly formal, you can take that inventory. I can know, when doing algebra (or deductions in the kinds of logic that are 'algebras') whether or not I have obeyed the rules. I can check at each step, because the allowed variations of operations are stated. But truly formal arguments work on models that need to be interpreted.

It is only if there are any elements of interpretation, that your argument holds water.

In that case, this is not wordplay, but is about the general weakness of linguistic conventions.

Wittgenstein would agree that proof is relative to a language game, and there is no global language game with a single standard of proof that will apply in all the others. (To my mind, that is what you have proven (if not very clearly...).)

But within the context of any game in question, even without formalism, proof is still well-defined. The current rules in a court of law, for instance, are understood. They are open to change, but reaching a closing move within the current rules still produces effective action. Relativism only matters until it doesn't anymore.

The question is not whether there is proof, but what kind of proof is expected by whom.

I can prove one earthworm plus one earthworm is two earthworms, but then what if one bites the other in half? Each half worm is a whole worm, the way worms work. Does math or biology win?

Do we throw out the mathematical principle as not proving anything because it has exceptions in other domains? No. It is not just a word game, it is a divergence of perspectives. (To put it in your terms, the math is overly precise and thus often wrong.)

Skepticism is a particularly useful perspective sometimes, and a pointless one at other times. But the fact that nothing can be proved to an absolute skeptic generally does not matter to anyone else, because humans tend work only inside some minimal set of games.

  • So you are saying that the regression argument (which I think is what I wanted to say actually) is basically a wordplay? – user 170039 Jun 8 '16 at 15:40
  • No. It is genuine, in its own right, as an illustration that classical logic itself is a game, not some magical grounding for all truth. But, as used, it is generally an evasion of the task at hand, rather than a contribution in good faith. It comes from two extreme position that do not fit together -- the realization that everything is vague, and a demand that truth be less vague than is therefore possible, when we are all pretty sure we know things.... – jobermark Jun 8 '16 at 16:35
  • Your paradox is real, because ultimately all definitions are circular, and require understandings that cannot be stated, and can only be partially communicated by examples and context or imposed by sheer elegance -- which means that for those very basic things we all actually have slightly different definitions. You are right that the more precise you want them, in general, the slipperier and more apparently circular they get. A cute dialog on this notion is math.berkeley.edu/~kpmann/Lakatos.pdf – jobermark Jun 8 '16 at 16:52
  • I think I misunderstood your post. Thanks for clarification. – user 170039 Jun 9 '16 at 3:15
  • However there is another thing that I think I need to clarify. You wrote that in "the context of any game in question, ..., proof is still well-defined."-it's true. But then don't we need to show that if we have 'proved' something, we actually have proved it, (i.e. in other words our 'proof' actually satisfies the definition of proof we have already given)? – user 170039 Jun 9 '16 at 3:37

How to assure that a logical conclusion is correct?

A logical conclusion is correct if and only if it adheres to the rules of logical reasoning. These rules are named valid syllogisms. One finds the table of all valid syllogism in textbooks on classical logic. Valid syllogism determine how a conclusion follows from two premiss.

E.g. Premiss 1 = "All rectangles have four equal angles", premiss 2: "All squares are rectangles", Conclusion = "All squares have four equal angles". This syllogism is named "modus barbara".

Hence a method to assure correct proofs is

  • to decompose a proof into a sequence of elementary steps and
  • to formalize each elementary step as a valid syllogism.
  • "A logical conclusion is correct if and only if it adheres to the rules of logical reasoning."- but how can we know for sure that it actually "adheres to the rules of logical reasoning"? – user 170039 Jun 8 '16 at 7:31
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    1. What exactly is a process of decomposing "a proof into a sequence of elementary steps"? 2. What is an "elementary step"? 3. What exactly is the process of formalizing each elementary step? 4. How can we be sure that our formalization of "each elementary step" is a valid syllogism? – user 170039 Jun 8 '16 at 7:40
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    2. The decomposition of a proof is a sequence of elementary steps.-but how can we be sure that our proof indeed can be decomposed into a sequence of elementary steps? Until and unless we can do that, we can't say for sure that our 'proof' is indeed a proof in the sense of 1. Furthermore, in a proof we are explicitly using only the valid syllogisms but how can we be sure that there is no implicit use of invalid syllogism? We need to ensure that too, right? – user 170039 Jun 8 '16 at 13:37
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    The problem is not in something like 'the square root of 2 is irrational' it is in something like 'this square wooden object has sides of length 2, so its diagonal is irrational'. It is valid that the square root of 2 is irrational, but upon application, it is not sound. The odds are quite high that I can adequately measure that diagonal and give it a rational number and not find my measure 'incorrect' because the deduction is 'correct' and contradicts it. The laws of logic are not complete enough to make that happen. The valid conclusion has failed to apply under interpretation. – jobermark Jun 8 '16 at 19:03
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    "Proof in general" is part of the OP. Restricting it to math would not be in order. Also, your answer does not restrict itself to math, nor are syllogisms used by most mathematicians. So this seems like an odd point to change all that. – jobermark Jun 8 '16 at 21:23

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