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Why doesn't mathematics collapse, though humans often make mistakes in proofs?

mathoverflow.net

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Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?

#11

Post by mitko »

Or does it...? Case in point, the fifth axiom of geometry, attributed to Euclid, that through a point there is exactly one line parallel to a given line has been shown to be unprovable, and Lobachevski and Rieman have built alternative geometric theories assuming that axiom wrong.

I find it hard to believe that Math is that different from physics or other sciences in its robustness.

Some advantages that math has had over other sciences is low resource consumption (pen and paper), easier reproducibility(thinking), and it has gotten a long head start of 2-3 thousand years.

Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?

#12

Post by nitwit005 »

The more boring answer correctness is testable. If I present a proof that two equations are equal, you don't have to trust me. You can try some values and see if they work, look for a contradiction, try to produce your own proof, or any other test you can think up.

Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?

#13

Post by diego »

Why would it even mean for mathematics to collapse? It's obvious that if a branch is based on rules that don't make sense, at some point that branch will break. But it only breaks to the root node, which is fine. But why would the whole edifice collapse?

Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?

#14

Post by brianberns »

Someone wrote: post #3

If 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…

The analogy I’d make is to bugs in software. Most software has bugs, but they’re only triggered in specific circumstances. Stay away from those cases and you’re usually fine. If you happen to hit a new bug, you might lose some data, but it’s not going to render the entire application useless.

Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?

#15

Post by booleandilemma »

Isn’t this like asking why society doesn’t collapse, when the software that drives it is filled with bugs?

My impression was: any result in modern mathematics critically depends on another result, and that result depends on some other result...

This is like how most large software systems depend on various 3rd party libraries.

I guess the answer for both questions is that the things we build, whether they be mathematics or software, work well enough most of the time that you don’t notice the problems unless you’re paying attention.

Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?

#16

Post by doubleunplussed »

mitko wrote: post #11

Or does it...? Case in point, the fifth axiom of geometry, attributed to Euclid, that through a point there is exactly one line parallel to a given line has been shown to be unprovable, and Lobachevski and Rieman have built alternative geometric theories assuming that axiom wrong. I find it hard to believe that Math is that different from physics or other sciences in its robustness. Some advantages that math has had…

Axioms shouldn't be provable (other than from themselves) - if the fifth axiom of geometry had turned out to be provable from something else, then it would be have just been renamed a theorem instead of an axiom.

Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?

#20

Post by bitL »

alephnan wrote: post #7
Someone wrote: post #3

If 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

Yeah, swapping out currently dominant logic with another one would redo the whole castle, like when a castle built in Gothic architecture is rebuilt into Baroque with some parts left intact. However, mainstream is so dominant and "better logics" either don't have sufficient manpower, are more complicated, are weaker or impractical, so that they never develop unless they solve something extraordinary quickly.
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