Author: tomr Format: TextAs I understand that there are at least two fundamental limits of the development of the mathematics:
1) Goedel incompeleteness theorems (or more clearly Church thesis) effectively says that there always will be unsolved mathematical problems (e.g. satisfiability of some formula or solution of some equation) that can not be solved by mechanical means but only by the creative thinking of some person. One can imagine that computer can be of help even for these problems, e.g. there is Connectionist approach to do symbolic computing by neural networks (or any other heuristic approach; there is journal "Connection Science" for this) and - as far as I understand - then neural networks in principle can generate solutions (and proofs) of these algorithmically unsolvable problems (although the developments are scarce in this field). There is even idea about computation beyond Turing limits (hypercomputation) that suggests that neural networks with irrational weights can go beyond Turing limits (as some physical machines, e.g. more powerful than the traditional non-relativistic low-energy quantum computers that relies on century old theories). So - this is fine and actually this is not a prohibitive limit.
2) The lack of soundness/completeness of some mathematical theories (higher-order logics) can be bigger problem: this means that the mathematical objects (ideas) can not be investigated by investigation of syntactical transformation of the words of some more or less formal language. The question is - what the other tools can be applied in such cases? Are those mathematical objects (ideas) are unavailable to the human reason and does mathematics stops here indeed? Is indeed the (experimental) physics necessary to uncover those objects (ideas)? Some years ago Hawking had to recognise the limits of theoretical physics arising from the Godel incompleteness theories. Does mathematicians should recognise their own limits?
I am sorry for such amateurish discussion and generally I am not thinking about these issues in my everyday activities but it would be interesting to know whether there are some trends in mathematics that are trying to overcome the mentioned limits. As was as I have read journals by the Association of Symbolic Logic (I guess, the best journals in mathematical logic and everything around it, e.g. reverse mathematics) or the Annals of Mathematics (I guess, the best journal in mathematic) then they are not especially concerned about such limits, they function quite happy within them. So - what is happening in this direction?
As I understand that there are at least two fundamental limits of the development of the mathematics:
1) Goedel incompeleteness theorems (or more clearly Church thesis) effectively says that there always will be unsolved mathematical problems (e.g. satisfiability of some formula or solution of some equation) that can not be solved by mechanical means but only by the creative thinking of some person. One can imagine that computer can be of help even for these problems, e.g. there is Connectionist approach to do symbolic computing by neural networks (or any other heuristic approach; there is journal "Connection Science" for this) and - as far as I understand - then neural networks in principle can generate solutions (and proofs) of these algorithmically unsolvable problems (although the developments are scarce in this field). There is even idea about computation beyond Turing limits (hypercomputation) that suggests that neural networks with irrational weights can go beyond Turing limits (as some physical machines, e.g. more powerful than the traditional non-relativistic low-energy quantum computers that relies on century old theories). So - this is fine and actually this is not a prohibitive limit.
2) The lack of soundness/completeness of some mathematical theories (higher-order logics) can be bigger problem: this means that the mathematical objects (ideas) can not be investigated by investigation of syntactical transformation of the words of some more or less formal language. The question is - what the other tools can be applied in such cases? Are those mathematical objects (ideas) are unavailable to the human reason and does mathematics stops here indeed? Is indeed the (experimental) physics necessary to uncover those objects (ideas)? Some years ago Hawking had to recognise the limits of theoretical physics arising from the Godel incompleteness theories. Does mathematicians should recognise their own limits?
I am sorry for such amateurish discussion and generally I am not thinking about these issues in my everyday activities but it would be interesting to know whether there are some trends in mathematics that are trying to overcome the mentioned limits. As was as I have read journals by the Association of Symbolic Logic (I guess, the best journals in mathematical logic and everything around it, e.g. reverse mathematics) or the Annals of Mathematics (I guess, the best journal in mathematic) then they are not especially concerned about such limits, they function quite happy within them. So - what is happening in this direction?