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What possible future mathematical methods are not considered rigorous math right now? [closed]
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In the same way that in the past the use of irrational numbers, calculus, and transfinite numbers were considered to be "not rigorous math" which contemporary mathematical constructs are causing controversy about not being "rigorous enough" but may have a shot to become an important part of the future mathematical canon?
soft-question math-history philosophy
edited Jul 30 at 12:58
Mauro ALLEGRANZA
60.6k346105
asked Jul 30 at 12:54
good_one
546
closed as unclear what you're asking by Peter, Did, Lord Shark the Unknown, user21820, MatheinBoulomenos Jul 30 at 17:37
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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up vote
3
down vote
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In the same way that in the past the use of irrational numbers, calculus, and transfinite numbers were considered to be "not rigorous math" which contemporary mathematical constructs are causing controversy about not being "rigorous enough" but may have a shot to become an important part of the future mathematical canon?
soft-question math-history philosophy
edited Jul 30 at 12:58
Mauro ALLEGRANZA
60.6k346105
asked Jul 30 at 12:54
good_one
546
closed as unclear what you're asking by Peter, Did, Lord Shark the Unknown, user21820, MatheinBoulomenos Jul 30 at 17:37
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
2
When were "transfinite numbera" not considered rigorous?
– Andrés E. Caicedo
Jul 30 at 13:11
This is probably opinion-based but I like it
– Robert Frost
Jul 30 at 15:10
@AndrésE.Caicedo I based it on excerpts from The Search for Certainty: A Philosophical Account of Foundations of Mathematics by Giaquinto p.30 that talks about the resistance of the mathematical community against transfinite numbers as numbers upon which arithmetic operations can be performed the then established division "between finite and infinite, the former being increasable by addition, the latter not." p.42
– good_one
Jul 30 at 17:42
I don't see where a lack of rigor shows up in that quote.
– Andrés E. Caicedo
Jul 30 at 17:56
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
In the same way that in the past the use of irrational numbers, calculus, and transfinite numbers were considered to be "not rigorous math" which contemporary mathematical constructs are causing controversy about not being "rigorous enough" but may have a shot to become an important part of the future mathematical canon?
soft-question math-history philosophy
edited Jul 30 at 12:58
Mauro ALLEGRANZA
60.6k346105
asked Jul 30 at 12:54
good_one
546
In the same way that in the past the use of irrational numbers, calculus, and transfinite numbers were considered to be "not rigorous math" which contemporary mathematical constructs are causing controversy about not being "rigorous enough" but may have a shot to become an important part of the future mathematical canon?
soft-question math-history philosophy
edited Jul 30 at 12:58
Mauro ALLEGRANZA
60.6k346105
asked Jul 30 at 12:54
good_one
546
edited Jul 30 at 12:58
Mauro ALLEGRANZA
60.6k346105
edited Jul 30 at 12:58
Mauro ALLEGRANZA
60.6k346105
edited Jul 30 at 12:58
Mauro ALLEGRANZA
60.6k346105
60.6k346105
asked Jul 30 at 12:54
good_one
546
asked Jul 30 at 12:54
good_one
546
asked Jul 30 at 12:54
good_one
546
546
closed as unclear what you're asking by Peter, Did, Lord Shark the Unknown, user21820, MatheinBoulomenos Jul 30 at 17:37
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Peter, Did, Lord Shark the Unknown, user21820, MatheinBoulomenos Jul 30 at 17:37
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
2
When were "transfinite numbera" not considered rigorous?
– Andrés E. Caicedo
Jul 30 at 13:11
This is probably opinion-based but I like it
– Robert Frost
Jul 30 at 15:10
@AndrésE.Caicedo I based it on excerpts from The Search for Certainty: A Philosophical Account of Foundations of Mathematics by Giaquinto p.30 that talks about the resistance of the mathematical community against transfinite numbers as numbers upon which arithmetic operations can be performed the then established division "between finite and infinite, the former being increasable by addition, the latter not." p.42
– good_one
Jul 30 at 17:42
I don't see where a lack of rigor shows up in that quote.
– Andrés E. Caicedo
Jul 30 at 17:56
add a comment |Â
2
When were "transfinite numbera" not considered rigorous?
– Andrés E. Caicedo
Jul 30 at 13:11
This is probably opinion-based but I like it
– Robert Frost
Jul 30 at 15:10
@AndrésE.Caicedo I based it on excerpts from The Search for Certainty: A Philosophical Account of Foundations of Mathematics by Giaquinto p.30 that talks about the resistance of the mathematical community against transfinite numbers as numbers upon which arithmetic operations can be performed the then established division "between finite and infinite, the former being increasable by addition, the latter not." p.42
– good_one
Jul 30 at 17:42
I don't see where a lack of rigor shows up in that quote.
– Andrés E. Caicedo
Jul 30 at 17:56
2
2
When were "transfinite numbera" not considered rigorous?
– Andrés E. Caicedo
Jul 30 at 13:11
When were "transfinite numbera" not considered rigorous?
– Andrés E. Caicedo
Jul 30 at 13:11
This is probably opinion-based but I like it
– Robert Frost
Jul 30 at 15:10
This is probably opinion-based but I like it
– Robert Frost
Jul 30 at 15:10
@AndrésE.Caicedo I based it on excerpts from The Search for Certainty: A Philosophical Account of Foundations of Mathematics by Giaquinto p.30 that talks about the resistance of the mathematical community against transfinite numbers as numbers upon which arithmetic operations can be performed the then established division "between finite and infinite, the former being increasable by addition, the latter not." p.42
– good_one
Jul 30 at 17:42
@AndrésE.Caicedo I based it on excerpts from The Search for Certainty: A Philosophical Account of Foundations of Mathematics by Giaquinto p.30 that talks about the resistance of the mathematical community against transfinite numbers as numbers upon which arithmetic operations can be performed the then established division "between finite and infinite, the former being increasable by addition, the latter not." p.42
– good_one
Jul 30 at 17:42
I don't see where a lack of rigor shows up in that quote.
– Andrés E. Caicedo
Jul 30 at 17:56
I don't see where a lack of rigor shows up in that quote.
– Andrés E. Caicedo
Jul 30 at 17:56
add a comment |Â
1 Answer
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In a talk called "Unclear Concepts" that I gave some years ago, I mentioned the following examples, which seem to fit your question.
(1) Feasible numbers. A natural number $n$ is said to be feasible if I can count up to it. So 1000 is feasible but $2^100$ is not. One expects some sort of induction principle to hold, at least when the number of "iterations" is feasible, i.e., if $P(0)$ and $(forall n),(P()to P(n+1))$, then all feasible $n$ should satisfy $P(n)$. With ordinary logic, this unfortunately lets you prove that $2^100$ is feasible; some version of linear or affine logic (omitting the contraction rule) seems more promising.
(2) Choice sequences. Brouwer introduced various versions of choice sequences in intuitionistic logic, but it seems difficult to provide a foundation for these in ordinary (classical) logic. Free choice sequences (where there are no restrictions on the choices) have been treated, but there seem to be problems if one is allowed to choose restrictions on future choices (including restrictions on future restrictions, etc.).
(3) The universe of all sets, i.e., the full cumulative hierarchy of sets, forming all subsets at each stage and iterating forever. Here "all" and "forever" are vague. Part of their meaning is well captured by the ZFC axioms (plus maybe some large cardinals), but the intuition seems to contain a lot more than the axioms, and it's not clear how to make that "lot more" rigorous.
answered Jul 30 at 13:21
Andreas Blass
47.4k348104
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
In a talk called "Unclear Concepts" that I gave some years ago, I mentioned the following examples, which seem to fit your question.
(1) Feasible numbers. A natural number $n$ is said to be feasible if I can count up to it. So 1000 is feasible but $2^100$ is not. One expects some sort of induction principle to hold, at least when the number of "iterations" is feasible, i.e., if $P(0)$ and $(forall n),(P()to P(n+1))$, then all feasible $n$ should satisfy $P(n)$. With ordinary logic, this unfortunately lets you prove that $2^100$ is feasible; some version of linear or affine logic (omitting the contraction rule) seems more promising.
(2) Choice sequences. Brouwer introduced various versions of choice sequences in intuitionistic logic, but it seems difficult to provide a foundation for these in ordinary (classical) logic. Free choice sequences (where there are no restrictions on the choices) have been treated, but there seem to be problems if one is allowed to choose restrictions on future choices (including restrictions on future restrictions, etc.).
(3) The universe of all sets, i.e., the full cumulative hierarchy of sets, forming all subsets at each stage and iterating forever. Here "all" and "forever" are vague. Part of their meaning is well captured by the ZFC axioms (plus maybe some large cardinals), but the intuition seems to contain a lot more than the axioms, and it's not clear how to make that "lot more" rigorous.
answered Jul 30 at 13:21
Andreas Blass
47.4k348104
add a comment |Â
up vote
3
down vote
In a talk called "Unclear Concepts" that I gave some years ago, I mentioned the following examples, which seem to fit your question.
(1) Feasible numbers. A natural number $n$ is said to be feasible if I can count up to it. So 1000 is feasible but $2^100$ is not. One expects some sort of induction principle to hold, at least when the number of "iterations" is feasible, i.e., if $P(0)$ and $(forall n),(P()to P(n+1))$, then all feasible $n$ should satisfy $P(n)$. With ordinary logic, this unfortunately lets you prove that $2^100$ is feasible; some version of linear or affine logic (omitting the contraction rule) seems more promising.
(2) Choice sequences. Brouwer introduced various versions of choice sequences in intuitionistic logic, but it seems difficult to provide a foundation for these in ordinary (classical) logic. Free choice sequences (where there are no restrictions on the choices) have been treated, but there seem to be problems if one is allowed to choose restrictions on future choices (including restrictions on future restrictions, etc.).
(3) The universe of all sets, i.e., the full cumulative hierarchy of sets, forming all subsets at each stage and iterating forever. Here "all" and "forever" are vague. Part of their meaning is well captured by the ZFC axioms (plus maybe some large cardinals), but the intuition seems to contain a lot more than the axioms, and it's not clear how to make that "lot more" rigorous.
answered Jul 30 at 13:21
Andreas Blass
47.4k348104
add a comment |Â
up vote
3
down vote
up vote
3
down vote
In a talk called "Unclear Concepts" that I gave some years ago, I mentioned the following examples, which seem to fit your question.
(1) Feasible numbers. A natural number $n$ is said to be feasible if I can count up to it. So 1000 is feasible but $2^100$ is not. One expects some sort of induction principle to hold, at least when the number of "iterations" is feasible, i.e., if $P(0)$ and $(forall n),(P()to P(n+1))$, then all feasible $n$ should satisfy $P(n)$. With ordinary logic, this unfortunately lets you prove that $2^100$ is feasible; some version of linear or affine logic (omitting the contraction rule) seems more promising.
(2) Choice sequences. Brouwer introduced various versions of choice sequences in intuitionistic logic, but it seems difficult to provide a foundation for these in ordinary (classical) logic. Free choice sequences (where there are no restrictions on the choices) have been treated, but there seem to be problems if one is allowed to choose restrictions on future choices (including restrictions on future restrictions, etc.).
(3) The universe of all sets, i.e., the full cumulative hierarchy of sets, forming all subsets at each stage and iterating forever. Here "all" and "forever" are vague. Part of their meaning is well captured by the ZFC axioms (plus maybe some large cardinals), but the intuition seems to contain a lot more than the axioms, and it's not clear how to make that "lot more" rigorous.
answered Jul 30 at 13:21
Andreas Blass
47.4k348104
In a talk called "Unclear Concepts" that I gave some years ago, I mentioned the following examples, which seem to fit your question.
(1) Feasible numbers. A natural number $n$ is said to be feasible if I can count up to it. So 1000 is feasible but $2^100$ is not. One expects some sort of induction principle to hold, at least when the number of "iterations" is feasible, i.e., if $P(0)$ and $(forall n),(P()to P(n+1))$, then all feasible $n$ should satisfy $P(n)$. With ordinary logic, this unfortunately lets you prove that $2^100$ is feasible; some version of linear or affine logic (omitting the contraction rule) seems more promising.
(2) Choice sequences. Brouwer introduced various versions of choice sequences in intuitionistic logic, but it seems difficult to provide a foundation for these in ordinary (classical) logic. Free choice sequences (where there are no restrictions on the choices) have been treated, but there seem to be problems if one is allowed to choose restrictions on future choices (including restrictions on future restrictions, etc.).
(3) The universe of all sets, i.e., the full cumulative hierarchy of sets, forming all subsets at each stage and iterating forever. Here "all" and "forever" are vague. Part of their meaning is well captured by the ZFC axioms (plus maybe some large cardinals), but the intuition seems to contain a lot more than the axioms, and it's not clear how to make that "lot more" rigorous.
answered Jul 30 at 13:21
Andreas Blass
47.4k348104
answered Jul 30 at 13:21
Andreas Blass
47.4k348104
answered Jul 30 at 13:21
Andreas Blass
47.4k348104
answered Jul 30 at 13:21
Andreas Blass
47.4k348104
47.4k348104
add a comment |Â
add a comment |Â
2
When were "transfinite numbera" not considered rigorous?
– Andrés E. Caicedo
Jul 30 at 13:11
This is probably opinion-based but I like it
– Robert Frost
Jul 30 at 15:10
@AndrésE.Caicedo I based it on excerpts from The Search for Certainty: A Philosophical Account of Foundations of Mathematics by Giaquinto p.30 that talks about the resistance of the mathematical community against transfinite numbers as numbers upon which arithmetic operations can be performed the then established division "between finite and infinite, the former being increasable by addition, the latter not." p.42
– good_one
Jul 30 at 17:42
I don't see where a lack of rigor shows up in that quote.
– Andrés E. Caicedo
Jul 30 at 17:56