heuristic thought in mathematical practice

[pyx]

how do you go about solving and integrating knowledge in mathematics on a conceptual/analytic level? what helps you resolve mathematical problems?

 

[CTB Dream]

Math rly brod trm ,hrstc can slv alrdy see hrstc algo see hrstc mthd, can slv prblty base no alws slv smtm slv smtm not,

if othr fld can see how aply math othr fld see cmpt fld many, also see ai ,this all math cncpt analtc

 

[pyx]

Math rly brod trm ,hrstc can slv alrdy see hrstc algo see hrstc mthd, can slv prblty base no alws slv smtm slv smtm not,

if othr fld can see how aply math othr fld see cmpt fld many, also see ai ,this all math cncpt analtc

i guess i was referring to mathematical problem-solving. and yeah math is too broad to really identify or constrain ourselves to a limited number of heuristic techniques, though i think there is a great deal of overlap (see Polya 1945).

 

[CTB Dream]

i guess i was referring to mathematical problem-solving. and yeah math is too broad to really identify or constrain ourselves to a limited number of heuristic techniques, though i think there is a great deal of overlap (see Polya 1945).

Same prblm slv same any, only diff lang type

 

[DarkRange55]

I sketch a picture illustrating the math problem.

- Mathematician

 

[CTB Dream]

I sketch a picture illustrating the math problem.

- Mathematician

ok mayb no undrstd this want way trnslt prblm? @lamargue

 

[pyx]

ok mayb no undrstd this want way trnslt prblm? @lamargue

i imagine that when dealing with abstract entities that pictures may not suffice.
but maybe a translation of the problem is accurate. in algebra we learn to solve more difficult equations by rote substitution, and thereby recognize patterns in equations and simplify them based on known identities (transformation rules). this is the same for most of calculus. it's different when you are trying to model a process with these known transformation rules, however.
when we arrive at analysis, we are introduced to the rigorous component of mathematical problem solving. the goal is to prove to a skeptical reader that your proof is correct, and you must reason in accordance with a set of pre-defined rules, i.e axioms, lemmas, etc., [these are incorporated within the proof] all of which to some degree lend to their own transformation rules; the difference is that it requires more substantive steps in order to piece together these elements. in computational analysis, i imagine that algorithms are used here, though in a far more rigorous way: the assumptions we carry are to a computer the constraints of the arithmetical model (i'm not a computer scientist). but that applies generally to numerical problems, where the heuristic chain of reasoning consist mainly of substitution chains. we don't really rely on computers in differential geometry.

and this sort of thinking crops up in set theory, ordinal arithmetic, model theory, etc., and sets the foundation for rigorous thinking.

when we transition into abstract algebra this becomes even more complicated. the limits of your reasoning are truly tested in this subject. if you arrived here, you would i imagine be at what terence tao called the post-rigorous stage. i've yet to arrive there myself as i still consider myself a novice, but you get the idea.

i wonder if this component of analysis can be used to reason clearly. for instance, modeling our assumptions as a logical entity may yield further results which in turn helps us to understand the bounds of the problem, if you will. it is a form of specification which i think may provide a great deal of use in higher mathematics. our brains are limited by our capacity to generate all the conditions of an arithmetic, but i think modeling our intuitions (which itself does not require that we establish all the rules of our system) would almost be a prerequisite in higher mathematics.

i really wonder how higher mathematicians think. maybe i haven't developed the intuition yet, idk

 

[DarkRange55]

I

i imagine that when dealing with abstract entities that pictures may not suffice.
but maybe a translation of the problem is accurate. in algebra we learn to solve more difficult equations by rote substitution, and thereby recognize patterns in equations and simplify them based on known identities (transformation rules). this is the same for most of calculus. it's different when you are trying to model a process with these known transformation rules, however.
when we arrive at analysis, we are introduced to the rigorous component of mathematical problem solving. the goal is to prove to a skeptical reader that your proof is correct, and you must reason in accordance with a set of pre-defined rules, i.e axioms, lemmas, etc., [these are incorporated within the proof] all of which to some degree lend to their own transformation rules; the difference is that it requires more substantive steps in order to piece together these elements. in computational analysis, i imagine that algorithms are used here, though in a far more rigorous way: the assumptions we carry are to a computer the constraints of the arithmetical model (i'm not a computer scientist). but that applies generally to numerical problems, where the heuristic chain of reasoning consist mainly of substitution chains. we don't really rely on computers in differential geometry.

and this sort of thinking crops up in set theory, ordinal arithmetic, model theory, etc., and sets the foundation for rigorous thinking.

when we transition into abstract algebra this becomes even more complicated. the limits of your reasoning are truly tested in this subject. if you arrived here, you would i imagine be at what terence tao called the post-rigorous stage. i've yet to arrive there myself as i still consider myself a novice, but you get the idea.

i wonder if this component of analysis can be used to reason clearly. for instance, modeling our assumptions as a logical entity may yield further results which in turn helps us to understand the bounds of the problem, if you will. it is a form of specification which i think may provide a great deal of use in higher mathematics. our brains are limited by our capacity to generate all the conditions of an arithmetic, but i think modeling our intuitions (which itself does not require that we establish all the rules of our system) would almost be a prerequisite in higher mathematics.

i really wonder how higher mathematicians think. maybe i haven't developed the intuition yet, idk

I got a phd in mathematical physics but currently work in finance. I will tell you this: Physics is concerned with the universe and what it's doing. Math is often concerned with its own self importance (math for math's sake). People that go into math are just generally people that like puzzles and stuff. I have a step-cousin who is one of the top researchers in the field of topology. He is a great example of this. Whereas I want to know the big questions and the philosophy of things and whats going on rather than just playing with some numbers.

 

[pyx]

I

I got a phd in mathematical physics but currently work in finance. I will tell you this: Physics is concerned with the universe and what it's doing. Math is often concerned with its own self importance (math for math's sake). People that go into math are just generally people that like puzzles and stuff. I have a step-cousin who is one of the top researchers in the field of topology. He is a great example of this. Whereas I want to know the big questions and the philosophy of things and whats going on rather than just playing with some numbers.

you're probably right. physics consists in providing adequate descriptions of the world.
but surely mathematicians also want to understand how the world works? or maybe that's a job for philosophers of mathematics. i'm not sure. there are polymaths like von neumann after all who believe:
"As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired from ideas coming from "reality", it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l'art pour l'art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste."

 

[DarkRange55]

you're probably right. physics consists in providing adequate descriptions of the world.
but surely mathematicians also want to understand how the world works? or maybe that's a job for philosophers of mathematics. i'm not sure. there are polymaths like von neumann after all who believe:
"As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired from ideas coming from "reality", it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l'art pour l'art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste."

Math can have proofs because it does not have to deal with the reality of the universe, so its assumptions don't have external dependencies. You can prove something within a closed system of assumptions in science (basically reducing that part of science to mathematics), but you can't really be sure that your assumptions reflect the way the universe works. Proof should only be used in a system where the axioms are clearly stated. In science, we cannot be sure that the axioms properly reflect reality, so we cannot really prove anything once we take reality into account.

This universe appears to obey principles that can be expressed using mathematics, so I look at mathematics as a tool/language for describing what the universe does. It is our best shot at a universal language among technologically advanced species.

"philosophy" meant understanding why, rather than just what. Philosophy deals with things like the meaning of truth and the meaning of proof, which are applied in mathematics.

 

[pyx]

Math can have proofs because it does not have to deal with the reality of the universe, so its assumptions don't have external dependencies. You can prove something within a closed system of assumptions in science (basically reducing that part of science to mathematics), but you can't really be sure that your assumptions reflect the way the universe works. Proof should only be used in a system where the axioms are clearly stated. In science, we cannot be sure that the axioms properly reflect reality, so we cannot really prove anything once we take reality into account.

This universe appears to obey principles that can be expressed using mathematics, so I look at mathematics as a tool/language for describing what the universe does. It is our best shot at a universal language among technologically advanced species.

"philosophy" meant understanding why, rather than just what. Philosophy deals with things like the meaning of truth and the meaning of proof, which are applied in mathematics.

i imagine that philosophy is really quite distinct from actual mathematical practice, yet the converse is not true. at least in analytic philosophy. mathematicians don't really consider the ontology of numerical systems or theory appraisal, for instance. i doubt mathematicians really care about this, at least while conducting mathematical research.

a physicist might care about theory appraisal, though.

 

[DarkRange55]

i imagine that philosophy is really quite distinct from actual mathematical practice, yet the converse is not true. at least in analytic philosophy. mathematicians don't really consider the ontology of numerical systems or theory appraisal, for instance. i doubt mathematicians really care about this, at least while conducting mathematical research.

a physicist might care about theory appraisal, though.

Philosophy still has a place at the edge of science.


Biology stands on the science of chemistry which in turn stands on atomic physics which stands on nuclear physics which stands on particle physics which stands on quantum mechanics which may stand on holography and quantum gravity.
(Basically, yes. But there's far more complexity between biology and chemistry than there is between atomic physics and particle physics...)

 

[DarkRange55]

i imagine that philosophy is really quite distinct from actual mathematical practice, yet the converse is not true. at least in analytic philosophy. mathematicians don't really consider the ontology of numerical systems or theory appraisal, for instance. i doubt mathematicians really care about this, at least while conducting mathematical research.

a physicist might care about theory appraisal, though.

rebuilding numbers from set theory?

 

[CTB Dream]

i imagine that when dealing with abstract entities that pictures may not suffice.
but maybe a translation of the problem is accurate. in algebra we learn to solve more difficult equations by rote substitution, and thereby recognize patterns in equations and simplify them based on known identities (transformation rules). this is the same for most of calculus. it's different when you are trying to model a process with these known transformation rules, however.
when we arrive at analysis, we are introduced to the rigorous component of mathematical problem solving. the goal is to prove to a skeptical reader that your proof is correct, and you must reason in accordance with a set of pre-defined rules, i.e axioms, lemmas, etc., [these are incorporated within the proof] all of which to some degree lend to their own transformation rules; the difference is that it requires more substantive steps in order to piece together these elements. in computational analysis, i imagine that algorithms are used here, though in a far more rigorous way: the assumptions we carry are to a computer the constraints of the arithmetical model (i'm not a computer scientist). but that applies generally to numerical problems, where the heuristic chain of reasoning consist mainly of substitution chains. we don't really rely on computers in differential geometry.

and this sort of thinking crops up in set theory, ordinal arithmetic, model theory, etc., and sets the foundation for rigorous thinking.

when we transition into abstract algebra this becomes even more complicated. the limits of your reasoning are truly tested in this subject. if you arrived here, you would i imagine be at what terence tao called the post-rigorous stage. i've yet to arrive there myself as i still consider myself a novice, but you get the idea.

i wonder if this component of analysis can be used to reason clearly. for instance, modeling our assumptions as a logical entity may yield further results which in turn helps us to understand the bounds of the problem, if you will. it is a form of specification which i think may provide a great deal of use in higher mathematics. our brains are limited by our capacity to generate all the conditions of an arithmetic, but i think modeling our intuitions (which itself does not require that we establish all the rules of our system) would almost be a prerequisite in higher mathematics.

i really wonder how higher mathematicians think. maybe i haven't developed the intuition yet, idk

See this all lang mthd etc prblm, have way can trnslt prblm smtm wrk smtm not, but can crtv do any imgn can do pic can imgn music story etc aby posbl wrk dpnde wat do,

say also human also lmt exmp this no posbl imgn v big nmbr

 

[pyx]

See this all lang mthd etc prblm, have way can trnslt prblm smtm wrk smtm not, but can crtv do any imgn can do pic can imgn music story etc aby posbl wrk dpnde wat do,

say also human also lmt exmp this no posbl imgn v big nmbr

yes and i think that's why heuristics are so important. mathematicians often have very good working memory, but even this has its limitations. a lot of contractive definitions (definitions which contain definitional chains of varying robustness) help to reduce the cognitive workload of the person handling them

rebuilding numbers from set theory?

a lot of mathematicians i imagine will learn this, as it's required for developing a really rigorous working mentality, and thus a more refined understanding of the boundaries of arithmetic (as developed in ordinal arithmetic and inductive proof techniques), though i think a philosopher would try to ask why these things work the way they do i.e how these abstract entities can be consistent, independent of our symbolism. Carnap distinguished this as working outside of the framework of the language (ontological questions), which is unnecessary for the mathematician

 

[DarkRange55]

yes and i think that's why heuristics are so important. mathematicians often have very good working memory, but even this has its limitations. a lot of contractive definitions (definitions which contain definitional chains of varying robustness) help to reduce the cognitive workload of the person handling them


a lot of mathematicians i imagine will learn this, as it's required for developing a really rigorous working mentality, and thus a more refined understanding of the boundaries of arithmetic (as developed in ordinal arithmetic and inductive proof techniques), though i think a philosopher would try to ask why these things work the way they do i.e how these abstract entities can be consistent, independent of our symbolism. Carnap distinguished this as working outside of the framework of the language (ontological questions), which is unnecessary for the mathematician

When hypotheses of "why" become testable, they become part of math/science.

 

[GuessWhosBack]

I think you would enjoy the field of mathematical logic, which treats notions such as proof in a rigorous manner. I can suggest books. You might actually be already familiar since you mentioned model theory.

With that being said, mathematical intuition, to me, is just an attempt to find results in the vast sea of provable statements, gone right. I'm not sure how mathematicians do it, but do keep in mind that sometimes it takes literal years for one result to get proven.