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в результате обмена мнений я поняла, что выяснять у математика, даже у профессора, вопрос об онтологическом статусе математических структур, это все равно что спрашивать у рыбы "мокро ли ей там под водой" - они там плавают как рыба в воде, а какое снаружи - это для пожалуй даже них смысла не имеет
philosopher:
It seemed to me that you consider the human intelligence and creativity to be the source of mathematical scientific knowledge. That is: we get understanding of the various regularities of the universe, when the mathematical representation of the processes in question are constructed according to the "mathematical rules of the game". Correct me if my impression is wrong.
Along with this view, the is an alternative idea, which considers the mathematical structures, formulas as some kind of entities embedded in the fabric of the universe, and thus the mathematicians are "just" recover the existing structures = formulas, like opening a can, or cracking the code.
I have difficulty in formulating the latter position, as obviously I think that the former is the case - humans (individuals and groups of mathematicians) are constructing knowledge in the forms of mathematical equations. Math is itself an evolving science - revisiting and reconstructing itself, correcting errors and adding more details, - this fact in my view demonstrates that it takes a human genius to recreate the universe in a form of code of the universe.
I would very much appreciate to learn your position and argumentation on the idea about the source of mathematical knowledge.
mathematician:
I actually believe that pure mathematicians do not invent new theories and formulas. These theories already exist. Pure mathematicians find them. The formulas
are already there. We need to find them.
Our insistence on proofs has to do with seeking absolute certainty. There are many examples of phenomena which hold for the first 100….even thousands of instances….but which
then break down miserably afterwards. Without proofs pure mathematicians don’t believe claims for this reason.
Our culture is very different than what one finds in theoretical physics (where this level of rigor is not expected). There are trade offs…
Theoretical physicists make many more mistakes as a result. However, progress in pure mathematics often seems to proceed at a glacial pace.
Best wises,
philosopher:
I really appreciate your response! Could you just clarify one more moment for me please? in attempt to verify the hypothesis about the formulas being already out there, am I missing something?
I'm avoiding the questions like "who might have programmed the universe", but my question is rather about the principles of constructing knowledge. Ramanujan's formulas would be lost unless some living humans found these and start reconstructing the links and missing sections and contributing the corrected results to the bigger body of knowledge. For me mathematics, theories and equations are about the humans inner creativity, humans create new and discard false theories and formulas, while the universe out there is always the same.
If the theories and formulas are "already there" and the mathematicians are only finding them, then we have to admit that both the true and erroneous theories and formulas are there. And if so, mathematicians still need to invent/ construct a principle for distinguishing the erroneous formulas and theories from those which are true (those which do and do not correspond to the regularities they try to describe), or the math professionals have to define the limited areas of applicability of specific formulas, some of which, as you say, may be true for the first 100 instances and wrong for the rest of the infinity of instances.
Thus, on one hand, if the principle of verification was invented by humans and for the humans to feel some certainty, to be able to select from all the formulas out there only those which humans feel good about, then this principle was not "out there"?
Or, on the other hand, if this principle of verification has always been out there among the theories and formulas, rather than being invented by the humans-mathematicians, then we have to admit an infinite number of true and false formulas mixed with an infinite number of different criteria for certainty. Is this the case?
mathematician:
I don’t consider wrong formulas as part of the fabric of the universe. These statements are simply wrong. The correct theorems are the ones which already exist….we simply need to find them.
However, we must prove them to confirm that they are indeed correct (instead of being wrong).
I actually also believe that the proofs already exist. The proofs are connections between theorems and correct formulas which hold the body of mathematics together.
Where then is creativity? Seeing or finding unknown facts takes creativity and imagination. Most of us do not have creative powers. We only know how to carry out calculations
which eventually become routine.
Ideas which are now routine were creative insights when they were first understood (e.g. understanding the meaning of derivatives….or limits….etc).
philosopher:
It seemed to me that you consider the human intelligence and creativity to be the source of mathematical scientific knowledge. That is: we get understanding of the various regularities of the universe, when the mathematical representation of the processes in question are constructed according to the "mathematical rules of the game". Correct me if my impression is wrong.
Along with this view, the is an alternative idea, which considers the mathematical structures, formulas as some kind of entities embedded in the fabric of the universe, and thus the mathematicians are "just" recover the existing structures = formulas, like opening a can, or cracking the code.
I have difficulty in formulating the latter position, as obviously I think that the former is the case - humans (individuals and groups of mathematicians) are constructing knowledge in the forms of mathematical equations. Math is itself an evolving science - revisiting and reconstructing itself, correcting errors and adding more details, - this fact in my view demonstrates that it takes a human genius to recreate the universe in a form of code of the universe.
I would very much appreciate to learn your position and argumentation on the idea about the source of mathematical knowledge.
mathematician:
I actually believe that pure mathematicians do not invent new theories and formulas. These theories already exist. Pure mathematicians find them. The formulas
are already there. We need to find them.
Our insistence on proofs has to do with seeking absolute certainty. There are many examples of phenomena which hold for the first 100….even thousands of instances….but which
then break down miserably afterwards. Without proofs pure mathematicians don’t believe claims for this reason.
Our culture is very different than what one finds in theoretical physics (where this level of rigor is not expected). There are trade offs…
Theoretical physicists make many more mistakes as a result. However, progress in pure mathematics often seems to proceed at a glacial pace.
Best wises,
philosopher:
I really appreciate your response! Could you just clarify one more moment for me please? in attempt to verify the hypothesis about the formulas being already out there, am I missing something?
I'm avoiding the questions like "who might have programmed the universe", but my question is rather about the principles of constructing knowledge. Ramanujan's formulas would be lost unless some living humans found these and start reconstructing the links and missing sections and contributing the corrected results to the bigger body of knowledge. For me mathematics, theories and equations are about the humans inner creativity, humans create new and discard false theories and formulas, while the universe out there is always the same.
If the theories and formulas are "already there" and the mathematicians are only finding them, then we have to admit that both the true and erroneous theories and formulas are there. And if so, mathematicians still need to invent/ construct a principle for distinguishing the erroneous formulas and theories from those which are true (those which do and do not correspond to the regularities they try to describe), or the math professionals have to define the limited areas of applicability of specific formulas, some of which, as you say, may be true for the first 100 instances and wrong for the rest of the infinity of instances.
Thus, on one hand, if the principle of verification was invented by humans and for the humans to feel some certainty, to be able to select from all the formulas out there only those which humans feel good about, then this principle was not "out there"?
Or, on the other hand, if this principle of verification has always been out there among the theories and formulas, rather than being invented by the humans-mathematicians, then we have to admit an infinite number of true and false formulas mixed with an infinite number of different criteria for certainty. Is this the case?
mathematician:
I don’t consider wrong formulas as part of the fabric of the universe. These statements are simply wrong. The correct theorems are the ones which already exist….we simply need to find them.
However, we must prove them to confirm that they are indeed correct (instead of being wrong).
I actually also believe that the proofs already exist. The proofs are connections between theorems and correct formulas which hold the body of mathematics together.
Where then is creativity? Seeing or finding unknown facts takes creativity and imagination. Most of us do not have creative powers. We only know how to carry out calculations
which eventually become routine.
Ideas which are now routine were creative insights when they were first understood (e.g. understanding the meaning of derivatives….or limits….etc).