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Knowledge is Good

October 6, 2020


Science is good too

Emil Faber is the pretend founder of the pretend Faber College. The 1978 movie Animal House starts with a close-up of Faber’s statue, which has the inscription, Knowledge Is Good.

Today, Ken and I thought we might talk about knowledge, science, mathematics, proofs, and more.

The phrase on Faber’s pedestal is meant to be a joke, as is the subtitle we added saying the same about science. But there is some truth to both of them. From the cause of climate change to the best response to the current pandemic to sports predictions there is much interest in science. Science is good, indeed.

Science

What is science and what are methods of creating knowledge via science? There is a whole world on the philosophy of science. The central questions are: What is science? What methods are used to create new science? Is science good?—just kidding.

We are not experts on the philosophy of science. But there seem to be three main ways to create scientific knowledge.

{\bullet } Experiments: This is the classic one. Think about the testing of a candidate vaccine to stop the pandemic.

{\bullet } Computational Experiments: This is relatively new. Think computer simulations of how climate change is effected by the methods of creating energy—for example. wind vs. coal.

{\bullet } Mathematical Proofs: This is the one we focus on here at GLL. Think proofs that some algorithm works or that there is no algorithm that can work unless…

Mathematical Proofs

We are interested in creating knowledge via proving new theorems. This is how we try to create knowledge. Our science is based not on experiments and not on simulations but mostly on the theorem-proof method. Well not exactly. We do use experiments and simulations. For example, the field of quantum algorithms uses both of these.

However, math proofs are the basis of complexity theory. This means that we need to create proofs and then check that they are correct. The difficulty of checking a proof is based on who created them:

  • You did—checking your own work.

  • Someone else did—refereeing for a journal.

  • Someone in your class did—grading exams.

  • Some graduate student did—mentoring.

  • Someone on the web who claims a major result like {\mathsf{P < NP}} did—debugging.

  • And so on.

My Favorite Checking Method

My favorite tool for checking is this trick: Suppose that we have a proof {P} that demonstrates {A \implies X} is true. Sometimes it is possible to show that there is a proof {Q} that proves {A \implies Y} where:

  1. The proof {Q} is based on changing the claimed proof {P}.

  2. The proof {Q} demonstrates {A \implies Y}, and;

  3. The statement {Y} does not follow from {A}.

One way this commonly arises is when {P} as a proof did not use all of the assumptions in {A}. Thus {P} really proves more that {X} and it proves {Y}. But we note that {Y} is not a consequence of {A}.

For example, consider the Riemann hypothesis. Suppose that we claim that we have a proof that

\displaystyle  \sum_{n=1}^{\infty} \frac{1}{n^{s}} \neq 0

follows from the usual axioms of math plus {\Re(s) > 1/2}. Sounds great. But suppose this is based on an argument that assumes that

\displaystyle  \sum_{n=1}^{\infty} \frac{1}{n^{s}} = 0

and manipulates the summation, eventually yielding a contradiction, without using the condition {\Re(s) > 1/2}. This is a problem, since there are {s} with {\Re(s) = 1/2} so that the sum is zero. This is an example of the above method of checking.

A New Checking Method

From time to time claims are made of resolutions to famous conjectures. Think {\mathsf{P = NP}}. These claims have all been wrong to date. So most researchers are reluctant to take time to check any new claims. Why would you take the effort to try and find the bug that is likely there?

I wonder if there could be a method that is based on competition. For concreteness, suppose Alice and Bob are two researchers who both claim a resolution to the {\mathsf{P}} versus {\mathsf{NP}} problem. Alice has a lower bound argument that {\mathsf{P < NP}} and Bob has an upper bound that {\mathsf{P = NP}}. Could we have them play a “game”?

Give their papers to each other. Have them try to find a flaw in each other’s paper.

They are highly motivated. Could we argue that if they cannot find any flaw then we would be slightly more motivated to look at the papers?

This might work even if they both claim {\mathsf{P = NP}}. Ken and I, personally, have had more claims of {\mathsf{P = NP}} brought to our attention. Even in this case they would be highly motivated: the awards, the prizes, the praise will go to the one who is correct.

Possible Extensions

One difference in our situation from classic empirical science is the nature of gaps in knowledge. For example, one of the big current controversies in physics is over the existence of dark matter. The Wikipedia article we just linked seems to date mostly to years around 2012 when dark matter was more widely accepted than strikes us today (see also this and this). There are cases where two competing theories are incompatible yet the available data do not suffice to find a fault in either.

Whereas, with claimed proofs of incompatible statements, such as {\mathsf{P < NP}} and {\mathsf{P = NP}}, at least one must have a demonstrable error. The statements themselves may have barriers all the way up to undecidability, but that does not matter to judging the proffered proofs.

The method may be more applicable in life sciences where the gap is gathering sufficient field or lab observations. For a topical example, consider claims about the risk or safety of human gatherings amid the pandemic. One extreme is represented by the extraordinary claim, which is evidently quite excessive, that the Sturgis motorcycle rally in August led to over 250,000 Covid-19 cases. The other extreme would be analyses used to justify gatherings with minimal precautions. The extremes cannot coexist. The means to arbitrate between them are available in principle but require costly social effort for contact tracing and testing as well as resolving mathematical issues between epidemiological models.

Open Problems

What do you think of our new checking method? Should it be more widely employed for evaluating claims and hypotheses?

26 Comments leave one →
  1. PandaItchpress permalink
    October 7, 2020 5:48 am

    P=NP. Even if SAT has a quadratic time algorithm how can it help factor in a reasonable time? You guys are pretty much idiots.

    • rjlipton permalink*
      October 7, 2020 7:29 am

      Dear Pandaltchpress:

      Thanks for your kind comment. If SAT is quadratic you are right. The obvious encoding of factoring would not be too fast. But that does not mean that there could not be a better encoding. So you are raising the question: Say SAT is quadratic time. Then how fast can one solve factoring?

      Best and stay well

      Dick

      • PandaItchpress permalink
        October 8, 2020 12:22 am

        No I am mocking the entire field of complexity and all the skillless idiots who are working in it. It is pretty obvious p is np and it has nothing to do with mathematics or nature.

      • PandaItchpress permalink
        October 8, 2020 4:57 am

        To the question how fast should factoring be? Almost as fast as multiplication. If it goes through sat perhaps slightly slower yet not practical. I really think the entire culture is made of idiots. At least morons.

      • PandaItchpress permalink
        October 8, 2020 5:13 am

        Asking to check p is np proof by factoring is the highest order moronic advice

    • javaid aslam permalink
      October 7, 2020 6:05 pm

      Wonder if you overlooked the reducibility of the factoring problem?

  2. Angela Weiss permalink
    October 7, 2020 8:39 am

    Funny, I always thought that Math is Philosophy.
    Science, Philosohy, Arts are good?
    Do they have borders?
    If there are no pure and strongly defined borders in which transpassers are severely punished, also, it is good. Anyway, I want think I am a Philosopher.
    Science is Good, Science is God? It is a good god for it allows doubt in its paradigms and, after a doubt is cast in open land, the sacred book of Science can be rewritten. Not static, as Plato believed but dancing and reshaping with no beginning or end or pause to freeze in the time.

    • October 7, 2020 11:58 am

      Angela, I agree with your dynamic characterization of what it means to know. I’m not sure I’d characterize Plato’s view as static (at least he wasn’t), but he didn’t use the word “science”. *Scientia*, which generally means knowledge, is the root that has the most agreement for most folks most of the time. I encounter disagreement when people mean that science is physics, chemistry, and biology (mathematicians are questionable, and engineers don’t get any respect!) In any case, everyone always agrees about the precepts, pay attention, be intelligent, responsibly act. These, of course, underlie scientific method (methods that result in knowledge), and everyone agrees about general method in data collection, interpreted meaning, and verification (or falsification!) in judgment of fact or knowledge. These words combine process and product, operations and outcomes, that are your dynamic knowing. I would only add the Polya’s How to Solve It gives an explicit description, and adds the important step of considering similar problems. In any case, for the computer “scientists” (and we know there is no science without scientists) I would defer to Knuth’s foreword in A=B, that, “Science is what we understand well enough to explain to a computer. Art is everything else we do.”

  3. Frank Vega permalink
    October 7, 2020 9:34 am

    Good idea Dick. I have had many requests in research gate about checking P = NP proofs, since I have been involved in this topic in the recent years. I have denied these revisions because of lack of time and motivation. Indeed, we need motivation to check a paper about P vs NP after knowing for certain the paper has great chances of being flawed. Sometimes, the motivation is to find some bright idea between another flawed ones. In any case, a motivation must guide the reading of such kind of papers. Hence, the motivation is the key and certainly we would have a good motivation if we have some benefits from our review and what is better than being revised on own claims? As I said, I have been involved in P-NP issues and many flawed attempts have arisen in the road. In this way, I invite any researcher to play “Dick’s Game” on my last attempt:

    https://doi.org/10.5281/zenodo.3355776

    I hope we could play 😀 !!!
    Thanks in advance,
    Frank

    • Frank Vega permalink
      October 8, 2020 6:45 pm

      Few minutes ago, I have discovered a way to finish my proof of the Riemann Hypothesis. Here you are:

      https://doi.org/10.5281/zenodo.3648511

      If there any researcher who wants to play “Dick’s Game” on this attempt, I am ready to play!!!
      But, take into account that is only in the Robin’s inequality which is the arguments of this paper.

      I hope we could play 😀 !!!
      Thanks in advance,
      Frank

    • Frank Vega permalink
      October 9, 2020 4:19 pm

      Today, I finally made my proof of the Riemann Hypothesis as short as I can. This only has a proof of 5 pages. See it:

      https://doi.org/10.5281/zenodo.3648511

      In case of being correct, this might be the most beauty, short, simple and the easy to understand proof about of an outstanding problem ever in whole history.

      What do you think?
      Frank

      • anne chev permalink
        October 14, 2020 10:20 am

        Congratulation, you solved one of the millenium problems and won 1M$

      • Frank Vega permalink
        October 17, 2020 1:15 pm

        Thanks…

  4. October 7, 2020 2:00 pm

    Dear Dick (& Ken) —

    It’s the usual thing to say scientific inquiry involves a combination of deductive and inductive reasoning.  A slightly different, 3-stage model, going back to Aristotle and revived by C.S. Peirce, analyzes the process producing knowledge into abductive, deductive, and inductive phases.  Abductive inference is what we use to generate a hypothesis, deduction is used to derive its logical consequences, and inductive reasoning is how we test the hypothesis against experimental observations.

    Here a few thoughts toward the design of software platforms for integrating these three components of inquiry.  (Also research and teaching.)

    An Architecture for Inquiry • Building Computer Platforms for Discovery

    Regards,

    Jon

    • October 7, 2020 2:53 pm

      Jon, apologies in advance for any inappropriate intrusion here, but I seem to recall your work being published in a different venue. I thought it was a similar group, and there was (it seemed) a valuable social organization where a critique of your pedagogy was included?

      • October 7, 2020 3:24 pm

        Critique Is Good 😉

      • October 7, 2020 3:29 pm

        yeah, I definitely didn’t mean “bad”, just-wondering if you had it conveniently available.

  5. jemand permalink
    October 7, 2020 3:11 pm

    In my eyes science (in contrast to craft) is expressed by “critique” as the attempt to distinguish the possible from the impossible. But since possibility-results sell much better than impossibility-results you hear today (for instance in a typical talk on machine learning) much about success-stories and what researchers are able to do and not so much about limits and what they cannot do.

  6. javaid aslam permalink
    October 7, 2020 6:08 pm

    Dear Dick (& Ken) —
    This new checking method can certainly increase the “throughput” of the reviews; as opposed to just being trashed…

  7. Ivan Gusev permalink
    October 11, 2020 11:43 am

    It wouldn’t work for folks that have gaps in knowledge. If both don’t understand natural barrier they cannot fix sm1’s proof that has it. When I looked at RH I found out that it has a lot of equivalent reformulations. So, naturally, one can assume that if sm1 solves RH he could show that his proof is rewritable into a proof for another equivalent statement. I think it is great way to get credibility for amateurs if they can show that their tools don’t have flaws, when they can rewrite them and get the same result. If I would proof RH I would post a paper titled smth like “Fifteen proofs of RH(for all famous equivalent forms)”. It would be easier to verify, any competent researcher could just take a look at one proof to really understand it. What is the way of proving RH that could be the most comprehensive to you? I feel that before P≠NP I will have to deal with RH anyway. My path will be showing certain properties of RZF, giving hints what they could mean, establishing tools in random matrix theory, Hamiltonian Mechanics, Dynamical systems with chaos, Operator theory, Algebraic topology and Analytic Number theory, then stating that they do that and that, and then showing that they do that and that, then showing that RZF is a subset of a bigger thing that is formulated by my tools. Then making several theorems that “show” new math and place of RH in it. Then showing path back, how to get back from RZF to my math, showing that the path works two ways. RZF says smth fundamental to my theory and my theory says smth fundamental about RZF and they are both true and agree with each other. So, what are your suggestions, what kind of proof of RH you would like to see and in what equivalent forms?

    • Frank Vega permalink
      October 11, 2020 12:35 pm

      Hi Ivan,

      Would you like to check my proof of the Riemann Hypothesis?

      https://doi.org/10.5281/zenodo.3648511

      If you remove the Introduction and Conclusion sections there are only 4 pages of proof. You may say is too short, but this proof demands from me a lot of effort and a many nights without sleeping.

      It will be good to hear a second opinion about it,
      I hope so,
      Thanks in advance,
      Frank

      • Ivan Gusev permalink
        October 11, 2020 2:25 pm

        none of iterative constructions would work. Suppose that we write f(a→b) like iterative function that has hidden parametrs that it guesses on its own in very smart way to describe certain actual statement with all its properties, given parametr s f() holds true, given different parametr r it is false. Suppose we have sequence of iterations in relation to given parametrs, we need to show that for all n/inN we have f to be true. f(1)(a→b), f(2)(a→b) and so one. In our proof we need to show that all n are in s and not a single r in it. Now lets reshuffle our iterations, they have to produce the same result for all n but it will not. The reason is simple function f has it’s own hidden parametrs that are infinitly conected to each other, so you cannot extract them into countable infinity easily without mindwrecking math. It will make length(=Complexity) of your proof infinte in size or simply wrong. But there is another way, you can “guess” numbers that are related to summation of thier “effect” on each other. Do it “simply” and I will belive that you can break RSA algorithm with your bare mind. I would suggest you to learn math for your own selfeducation and to spend time on smth else. RH is not solvable if you don’t know the secret of its zeroes. I do know and it is pretty simple and elegant but if you don’t know the secret then it has infinite complexity to you, salami slicing will yield nothing, numerical plays are unlikely(like breaking RSA), analysis will always be too imprecise, or you will always fool yrslf. Trying to solve RH is the most meaningless endeavour for everyone. You will not beat it, it can only make you stronger if you please to tolerate all pain and grow.

      • Frank Vega permalink
        October 11, 2020 6:53 pm

        Thank you very much Ivan for your pieces of advice!!!

Trackbacks

  1. Knowledge Is Good – A2M1N
  2. Knowledge Is Good - GistTree
  3. Theme One • A Program Of Inquiry 19 | Inquiry Into Inquiry

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