Why doesn't mathematics collapse, though humans often make mistakes in proofs?
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Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#2"Once in a while, I like to indulge in an informative anecdote concerning the genesis of the proof. The criterion was found by the end of 1985; then I remained more than six months making circles around the « splitting tensor ». One nice day of August 1986, I woke up in a camp in Siena and I had got the proof: I therefore sat down and wrote a manuscript of 10 pages. One month later, while recopying this with my typewriter, I discovered that one of my lemmas about imperialism was wrong: no importance, I made another one! This illustrates the fact, neglected by the formalist ideology, that a proof is not the mere putting side by side of logical rules, it is a global perception: since I had found the concept of empire, I had my theorem and the faulty lemma was hardly more than a misprint."
- Jean-Yves Girard, The Blind Spot
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#3If builders of a huge castle place some stones incorrectly, does it collapse completely?
No. Such errors may be discovered soon and taken down, or stand for decades, sometimes with lots of stuff built on top of it, and sometime parts will collapse, but the overall structure of the castle is sound, and, sooner or later, errors will be corrected.
And of course, sometimes, somebody decides to start building an entire new wing to the castle using new rules, as, for example, happened when non-Euclidean geometry was discovered/invented.
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#4Mathematics is convex to errors
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#5I think it wise to be skeptical of any new result coming from a complicated proof until it has been proven multiple ways by multiple people. If multiple approaches are not easily attained, then passing the test of multiple counterexample attempts can at least provide some degree of confidence. If something new is significant, it tends to attract that kind of scrutiny pretty quickly. I think that is the saving grace of the mathematics community.
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#6Mathematics is convex to errors
Supposing that the definition of convexity that you're referring to is that of, for a set C with x_1 and x_2 being elements of C and for any scalar b between zero and one inclusively,
[b * x_1 + (1 - b) * x_2] always being a member of C,
I presume that you mean that errors, namely those kind of errors that are the subject of this thread's discussion, in mathematics will always symbolically be elements of some solution set of a finite number of linear equalities and inequalities (or the intersection of a finite number of halfspaces and hyperplanes).
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#7If builders of a huge castle place some stones incorrectly, does it collapse completely? No. Such errors may be discovered soon and taken down, or stand for decades, sometimes with lots of stuff built on top of it, and sometime parts will collapse, but the overall structure of the castle is sound, and, sooner or later, errors will be corrected. And of course, sometimes, somebody decides to start building an entire ne…
On the analogy of castles, I asked a professor whether Godel’s
Incompleteness theorems affects their work. She told me Mathematics is this great castle, where in one quadrant you have the number theorist, in another the topologists. Every decade or century, the logicians would come running up from the basement with results confronting the foundations of math, but everyone else continued with their routine
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#8If builders of a huge castle place some stones incorrectly, does it collapse completely? No. Such errors may be discovered soon and taken down, or stand for decades, sometimes with lots of stuff built on top of it, and sometime parts will collapse, but the overall structure of the castle is sound, and, sooner or later, errors will be corrected. And of course, sometimes, somebody decides to start building an entire ne…
On the analogy of castles, I asked a professor whether Godel’s Incompleteness theorems affects their work. She told me Mathematics is this great castle, where in one quadrant you have the number theorist, in another the topologists. Every decade or century, the logicians would come running up from the basement with results confronting the foundations of math, but everyone else continued with their routine
Perhaps the point is that the basement of a castle is also called a dungeon.
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#9Mathematics is convex to errors
Supposing that the definition of convexity that you're referring to is that of, for a set C with x_1 and x_2 being elements of C and for any scalar b between zero and one inclusively, [b * x_1 + (1 - b) * x_2] always being a member of C, I presume that you mean that errors, namely those kind of errors that are the subject of this thread's discussion, in mathematics will always symbolically be elements of some solutio…
It is informal and means that mistakes hurt less than non-mistakes help.
A hypertechnical article that gives the gist:
https://www.edge.org/conversation/nassim_nicholas_taleb-unde...
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#10The simpler and more elegant the proof, the less likely we are to commit an error