Can we be sure proofs have no errors?
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My current understanding is that work submitted to journals has mathematicians look over it for errors. Mathematics is deductive, yet with this being the burden of proof, how can we know for sure there have been no mistakes and we are demanding enough rigour in our burden of proof?
For example, in the 20th and 19th century I know there were some issues when calculus was not formalised rigorously enough, leading to contradictions. Also, when you go to higher levels mathematics, you find that your previous acceptable burden of proof was totally inadequate. I am concerned because, if we cannot be sure, then mathematics instead becomes very empirical - we use a certain burden of proof only because that has not led to any contradictions yet.
I read the comments in >implying we can discuss mathematics, title 'I recently came into an innocent-looking discussion, which leads me to ask the following: Do you think mathematics is a science? Why? Why not?'
I also read these articles, but haven't found a satisfactory answer yet. Thanks!
https://www.newscientist.com/article/mg19125661-400-mathematics-the-burden-of-proof/
https://thesociablesolipsist.com/2018/08/26/is-mathematics-empirical/
logic philosophy proof-theory
asked Sep 5 at 8:48
Ethan Horsfall
1726
add a comment |Â
up vote
0
down vote
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My current understanding is that work submitted to journals has mathematicians look over it for errors. Mathematics is deductive, yet with this being the burden of proof, how can we know for sure there have been no mistakes and we are demanding enough rigour in our burden of proof?
For example, in the 20th and 19th century I know there were some issues when calculus was not formalised rigorously enough, leading to contradictions. Also, when you go to higher levels mathematics, you find that your previous acceptable burden of proof was totally inadequate. I am concerned because, if we cannot be sure, then mathematics instead becomes very empirical - we use a certain burden of proof only because that has not led to any contradictions yet.
I read the comments in >implying we can discuss mathematics, title 'I recently came into an innocent-looking discussion, which leads me to ask the following: Do you think mathematics is a science? Why? Why not?'
I also read these articles, but haven't found a satisfactory answer yet. Thanks!
https://www.newscientist.com/article/mg19125661-400-mathematics-the-burden-of-proof/
https://thesociablesolipsist.com/2018/08/26/is-mathematics-empirical/
logic philosophy proof-theory
asked Sep 5 at 8:48
Ethan Horsfall
1726
Proof are published on journals/books: thus, they are review by many mathematicians. On the long term, this process will discover existing errors (if any). This "intersubjective" aspects of math is what ground its objectivity, and thus : YES, math is a science.
– Mauro ALLEGRANZA
Sep 5 at 8:51
2
We can use computers to check proof, but then again there may be errors in the code or bugs in the computer. In any case, a given proof, written the usual way will always be subject to doubt; but that does not make mathematics empirical or a science : in principle, there is a sure strategy to eliminate errors, which consists in translating everything to a formal proof (e.g. in natural deduction) and if you can't, it's not a proof. But that would be annoying and painstakingly long and complicated - but in principle we can do it
– Max
Sep 5 at 9:16
Surely a computer checking a proof would be a circular argument, as we have designed them and trust them on certain mathematical arguments, and this problem makes those mathematical arguments suspect
– Ethan Horsfall
Sep 5 at 9:45
@EthanHorsfall All logic is ultimately circular in the presence of sufficient skepticism; to even reason at the most basic level, we need to accept a certain threshold of confidence. The point is that relative to an incredibly microscopic commitment, the approach above is perfectly valid.
– Noah Schweber
Sep 5 at 17:58
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
My current understanding is that work submitted to journals has mathematicians look over it for errors. Mathematics is deductive, yet with this being the burden of proof, how can we know for sure there have been no mistakes and we are demanding enough rigour in our burden of proof?
For example, in the 20th and 19th century I know there were some issues when calculus was not formalised rigorously enough, leading to contradictions. Also, when you go to higher levels mathematics, you find that your previous acceptable burden of proof was totally inadequate. I am concerned because, if we cannot be sure, then mathematics instead becomes very empirical - we use a certain burden of proof only because that has not led to any contradictions yet.
I read the comments in >implying we can discuss mathematics, title 'I recently came into an innocent-looking discussion, which leads me to ask the following: Do you think mathematics is a science? Why? Why not?'
I also read these articles, but haven't found a satisfactory answer yet. Thanks!
https://www.newscientist.com/article/mg19125661-400-mathematics-the-burden-of-proof/
https://thesociablesolipsist.com/2018/08/26/is-mathematics-empirical/
logic philosophy proof-theory
asked Sep 5 at 8:48
Ethan Horsfall
1726
My current understanding is that work submitted to journals has mathematicians look over it for errors. Mathematics is deductive, yet with this being the burden of proof, how can we know for sure there have been no mistakes and we are demanding enough rigour in our burden of proof?
For example, in the 20th and 19th century I know there were some issues when calculus was not formalised rigorously enough, leading to contradictions. Also, when you go to higher levels mathematics, you find that your previous acceptable burden of proof was totally inadequate. I am concerned because, if we cannot be sure, then mathematics instead becomes very empirical - we use a certain burden of proof only because that has not led to any contradictions yet.
I read the comments in >implying we can discuss mathematics, title 'I recently came into an innocent-looking discussion, which leads me to ask the following: Do you think mathematics is a science? Why? Why not?'
I also read these articles, but haven't found a satisfactory answer yet. Thanks!
https://www.newscientist.com/article/mg19125661-400-mathematics-the-burden-of-proof/
https://thesociablesolipsist.com/2018/08/26/is-mathematics-empirical/
logic philosophy proof-theory
logic philosophy proof-theory
asked Sep 5 at 8:48
Ethan Horsfall
1726
asked Sep 5 at 8:48
Ethan Horsfall
1726
asked Sep 5 at 8:48
Ethan Horsfall
1726
asked Sep 5 at 8:48
Ethan Horsfall
1726
asked Sep 5 at 8:48
Ethan Horsfall
1726
1726
Proof are published on journals/books: thus, they are review by many mathematicians. On the long term, this process will discover existing errors (if any). This "intersubjective" aspects of math is what ground its objectivity, and thus : YES, math is a science.
– Mauro ALLEGRANZA
Sep 5 at 8:51
2
We can use computers to check proof, but then again there may be errors in the code or bugs in the computer. In any case, a given proof, written the usual way will always be subject to doubt; but that does not make mathematics empirical or a science : in principle, there is a sure strategy to eliminate errors, which consists in translating everything to a formal proof (e.g. in natural deduction) and if you can't, it's not a proof. But that would be annoying and painstakingly long and complicated - but in principle we can do it
– Max
Sep 5 at 9:16
Surely a computer checking a proof would be a circular argument, as we have designed them and trust them on certain mathematical arguments, and this problem makes those mathematical arguments suspect
– Ethan Horsfall
Sep 5 at 9:45
@EthanHorsfall All logic is ultimately circular in the presence of sufficient skepticism; to even reason at the most basic level, we need to accept a certain threshold of confidence. The point is that relative to an incredibly microscopic commitment, the approach above is perfectly valid.
– Noah Schweber
Sep 5 at 17:58
add a comment |Â
Proof are published on journals/books: thus, they are review by many mathematicians. On the long term, this process will discover existing errors (if any). This "intersubjective" aspects of math is what ground its objectivity, and thus : YES, math is a science.
– Mauro ALLEGRANZA
Sep 5 at 8:51
2
We can use computers to check proof, but then again there may be errors in the code or bugs in the computer. In any case, a given proof, written the usual way will always be subject to doubt; but that does not make mathematics empirical or a science : in principle, there is a sure strategy to eliminate errors, which consists in translating everything to a formal proof (e.g. in natural deduction) and if you can't, it's not a proof. But that would be annoying and painstakingly long and complicated - but in principle we can do it
– Max
Sep 5 at 9:16
Surely a computer checking a proof would be a circular argument, as we have designed them and trust them on certain mathematical arguments, and this problem makes those mathematical arguments suspect
– Ethan Horsfall
Sep 5 at 9:45
@EthanHorsfall All logic is ultimately circular in the presence of sufficient skepticism; to even reason at the most basic level, we need to accept a certain threshold of confidence. The point is that relative to an incredibly microscopic commitment, the approach above is perfectly valid.
– Noah Schweber
Sep 5 at 17:58
Proof are published on journals/books: thus, they are review by many mathematicians. On the long term, this process will discover existing errors (if any). This "intersubjective" aspects of math is what ground its objectivity, and thus : YES, math is a science.
– Mauro ALLEGRANZA
Sep 5 at 8:51
Proof are published on journals/books: thus, they are review by many mathematicians. On the long term, this process will discover existing errors (if any). This "intersubjective" aspects of math is what ground its objectivity, and thus : YES, math is a science.
– Mauro ALLEGRANZA
Sep 5 at 8:51
2
2
We can use computers to check proof, but then again there may be errors in the code or bugs in the computer. In any case, a given proof, written the usual way will always be subject to doubt; but that does not make mathematics empirical or a science : in principle, there is a sure strategy to eliminate errors, which consists in translating everything to a formal proof (e.g. in natural deduction) and if you can't, it's not a proof. But that would be annoying and painstakingly long and complicated - but in principle we can do it
– Max
Sep 5 at 9:16
We can use computers to check proof, but then again there may be errors in the code or bugs in the computer. In any case, a given proof, written the usual way will always be subject to doubt; but that does not make mathematics empirical or a science : in principle, there is a sure strategy to eliminate errors, which consists in translating everything to a formal proof (e.g. in natural deduction) and if you can't, it's not a proof. But that would be annoying and painstakingly long and complicated - but in principle we can do it
– Max
Sep 5 at 9:16
Surely a computer checking a proof would be a circular argument, as we have designed them and trust them on certain mathematical arguments, and this problem makes those mathematical arguments suspect
– Ethan Horsfall
Sep 5 at 9:45
Surely a computer checking a proof would be a circular argument, as we have designed them and trust them on certain mathematical arguments, and this problem makes those mathematical arguments suspect
– Ethan Horsfall
Sep 5 at 9:45
@EthanHorsfall All logic is ultimately circular in the presence of sufficient skepticism; to even reason at the most basic level, we need to accept a certain threshold of confidence. The point is that relative to an incredibly microscopic commitment, the approach above is perfectly valid.
– Noah Schweber
Sep 5 at 17:58
@EthanHorsfall All logic is ultimately circular in the presence of sufficient skepticism; to even reason at the most basic level, we need to accept a certain threshold of confidence. The point is that relative to an incredibly microscopic commitment, the approach above is perfectly valid.
– Noah Schweber
Sep 5 at 17:58
add a comment |Â
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Proof are published on journals/books: thus, they are review by many mathematicians. On the long term, this process will discover existing errors (if any). This "intersubjective" aspects of math is what ground its objectivity, and thus : YES, math is a science.
– Mauro ALLEGRANZA
Sep 5 at 8:51
2
We can use computers to check proof, but then again there may be errors in the code or bugs in the computer. In any case, a given proof, written the usual way will always be subject to doubt; but that does not make mathematics empirical or a science : in principle, there is a sure strategy to eliminate errors, which consists in translating everything to a formal proof (e.g. in natural deduction) and if you can't, it's not a proof. But that would be annoying and painstakingly long and complicated - but in principle we can do it
– Max
Sep 5 at 9:16
Surely a computer checking a proof would be a circular argument, as we have designed them and trust them on certain mathematical arguments, and this problem makes those mathematical arguments suspect
– Ethan Horsfall
Sep 5 at 9:45
@EthanHorsfall All logic is ultimately circular in the presence of sufficient skepticism; to even reason at the most basic level, we need to accept a certain threshold of confidence. The point is that relative to an incredibly microscopic commitment, the approach above is perfectly valid.
– Noah Schweber
Sep 5 at 17:58