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I was just pondering as a mathematics major, is there a particular instance where a mathematician's work doe NOT require agreements among peer scholars of mathematics to determine the quality of the knowledge being produced in mathematics?

I do think that mathematics does not require agreements to assess the quality of the knowledge being produced because the mathematics is a self evident and standing subject. But also I am curious on what the arguments may be because despite my already posted question I am still confused with the question:

"why wouldn't mathematics require consensus for determining the quality of knowledge?"

Can anyone give me an in depth argument regarding this topic?

Thank you in advance.

EDIT: For instance the quality of knowledge I mean in this particular case refers to how self - evident the logical statements being formed through axiomatic systems which are combined and used in various but appropriate ways. But what I am truly asking is, how can the quality of knowledge in mathematics not be determined by the agreement which comes along with it? In other words, how self evident can mathematics really be, to such an extent that it does not require consensus to determine its validity?

  • Obviously mathematics needs "agreement" with members of the math community. See Georg Cantor and Luitzen Egbertus Jan Brouwer. – Mauro ALLEGRANZA Nov 24 '18 at 9:51
  • "why wouldn't mathematics require consensus for determining the quality of knowledge?" See the post Is a proof still valid if only the author understands it ? – Mauro ALLEGRANZA Nov 24 '18 at 9:53
  • I still do not understand this notion or argument which support how mathematics doesn't require consensus to determine the quality of knowledge being produced.Is there like a good text ADDRESSING platonic realism where it purely supports mathematical truthfulness purely by the virtue of its existence? I do not get the argument posed in the links you have provided, I will appreciate some indepth explanation. – Aurora Borealis Nov 24 '18 at 12:02
  • What do you mean by 'quality of knowledge'? Do you mean whether or not the theorems being proven are true, or are useful, or are interesting, or something else? Right now your question is vague because it is not clear what exactly it is you are asking about. – Not_Here Nov 24 '18 at 14:12
  • I am trying to figure out if consensus is required to determine the quality of knowledge in mathematics. The quality of knowledge I am referring here refers to how true the knowledge is. Could you enlighten me with some support backing this idea? – Aurora Borealis Nov 24 '18 at 15:33
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To answer this question, one must ask: What is quality?

If quality is correctness, then all a work requires is to show beyond reasonable doubt that the relevant logical steps have been taken from the axioms to the given conclusions.

If quality is usefulness, then a work needs to satisfy the criteria for correctness, along with methods, premeses or conclusions which are relevant to future work, and as a result usually building on previous work.

If quality, on the other hand, is something like fame, then consensus is sufficient but not necessary, as there are example of (in)famous people like Miles Mathis using shoddy methods to prove such absurdities as "pi = 4".

Different understandings of quality lead to different answers to the question.

  • "What is quality?" -- Ah, Zen and the Art of Motorcycle Maintenance! – user4894 Nov 24 '18 at 18:38
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Plato's notion of anamnesis may not apply to much of the world. But in mathematics, his chosen example in the Theatetus, it seems right. Logic does not require consensus to determine its validity because it is an inborn feature of human beings. Even if we are not trained in any kind of formal reasoning, at a given age and level of intelligence, we have a predictable reaction to contradictions and impossibilities. The most basic kinds of math seem more like remembering than learning.

Mathematics elaborates logic, and seeks out similar things that we will naturally all agree upon (e.g. a straight line as we naturally imagine it, until we think about it too seriously, is infinite and infinitely divisible.)

It can be wrong about what those are, but once it has found one, it can reasonably combine it with all the others it has already collected.

In the history of mathematics, ideas really do become mathematics by consensus. Take infinitesimals. They were basic tools for early physics. But then we found the idea was not thoroughly shared, to the degree we got different results from different people when we combined infinitesimal proofs with other known math. They got thrown away and worked around, with the project of 'arithmetizing' analysis. But once we played with them enough, they got reconstructed in different forms until they took a few forms that can be agreed upon.

Yet once they are there, they are there. L/os theorem, Ultraproducts and 'External' numbers are never going away, they seem to really capture this intuition and to be formally stable enough to combine it with the rest of math. It turns out that the intuition was real and stable, despite prior consensus this was not the case.

On the other hand, mathematical Platonism does fail. There is good evidence that mathematics already contains intuitions that cannot be combined. Russel's theorem suggests that we need to better refine our notions of self-reference, universality or negation. We have silly workarounds, but in the end, the contradiction is really there. When people get close to those boundaries, like the people working in ordinal theory, there is a real necessity for consensus, and a competition between interpretations. It is possible that some of what is accepted now really does lie too close to the boundaries where things break down and within the area ruled out by Russel's paradox and the resulting notion of 'proper classes'. Whether to consider it math sometimes remains a judgement call. We will know only when the building upon it proves useful or falls apart.

Likewise, there are competing relative concepts in math that are in direct competition. Our definition of measuring space and the set-theoretic Axiom of Choice are at odds. Together they give us Tarski's paradox. Whether that is a problem or just a quirk of mathematical reality is not an issue where there is consensus. Folks who think it is a problem have proven the axiom is largly dispensable, and not very much ultimately depends upon it. So they look askance at its continued use. Others have even proposed axioms that seem more useful and rule out this possibility, like the Axiom of Determinacy.

So your overall impression that all of mathematics is free of subjective judgement is not real. But large tracts of it are safe, without seeking consensus, because they are built upon inborn reactions and filtered by rather harsh standards before we start building on them.

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To answer the first question: "is there a particular instance where a mathematician's work does NOT require agreements among peer scholars of mathematics to determine the quality of the knowledge being produced in mathematics?"

Yes, there are some simple results where outsiders are much more reliable than professionally blinkered mathematicians. For instance a normal person would never believe in finished infinity or in undefinable real numbers or in the possibility to separate uncountably many parts by countably many parts in a linear continuum.

"I do think that mathematics does not require agreements to assess the quality of the knowledge being produced because the mathematics is a self evident and standing subject."

This is wrong. All of transfinite set theory and most of topology is only existing because of professional consensus to maintain it against self evident facts like "every real number has a finite definition".

To name only one striking example:

Axel Harnack, a pioneer of measure theory, has proved in 1885 that a countable number of intervals removed from the real axis leaves a countable number of intervals in the complement.

This shows either that the measure of the real axis is zero or that set theory, in particular the notion of countability (that cannot be proved without assuming finished infinity) is wrong.

But Émile Borel "proved" that the complement of a countable set of intervals can be an uncountable set of (separated degenerate) intervals.

That "proof" is obviously false and would even disprove set theory: Every finite number of removed parts leaves a finite number of separated parts in the complement. Should the number of parts in the complement "explode" as soon as the number of removed parts is infinite, then the same could happen in Cantor's diagonal argument: The list could "explode" too, when being infinite. In the same way also all other proofs of uncountable sets would fail.

Further, if

(1) removal of countably many parts leaves uncountably many separated parts, then

(2) removal of uncountably many parts leaves countably many separated parts.

(2) follows by simply symmetry: removing what has been left in (1). But when we stop in (2) after having removed only countably many parts, then we have separated more parts than after finishing.

This argument can only be circumvented by refusing symmetry considerations and further by assuming that single points in the real axis can exist without connection to their neighbourhood, i.e., by claiming that these points are points of the complement but are neither in the limit nor in the interior of the complement. That is not at all self evident - it is a fraud. For more technical details see https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 326f.

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    You're showing a nearly complete lack of understanding of mathematics here. It isn't formally connected to the real world, and what is self-evident in the real world is irrelevant to mathematics. For example, "every real number has a finite definition" is clearly wrong mathematically, since the reals are uncountable and finite definitions are countable. In a practical sense, there is very little use for a real that doesn't have a finite definition, but that doesn't mean the math is wrong. – David Thornley Jan 7 at 22:57
  • @DavidThornley Ibrahim is one of Wolfgang Mückenheim's sockpuppets, so you might drop the word "here" in the first sentence :-) Google is your friend. – Uwe Jan 8 at 0:43
  • @David Thornley: Of course I cannot understand a self-contradictory theory. "All real numbers can be well-ordered." This has been "proved" by Zermelo when he was not yet aware of the fact that uncountability implies undefinability. If you believe that undefinable "real" numbers can be well-ordered, then you show a complete lack of consisten thinking. By what property would you proceed? – Ibrahim Abd el Faruk-Shaik Jan 8 at 8:05
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    (If you are here to preach instead of to argue, go find some religious site. This one is about philosophy.) – jobermark Jan 10 at 9:49
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    This is utter nonsense, and if people are deleting your stuff, you have earned it. – jobermark Jan 10 at 17:47

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