If mathematics is indispensably part of our best scientific theories, does the scientific confirmation of those theories provide a reason to accept the existence claims of mathematics?
There is an aspect of the indispensability question that has always struck me as conspicuously absent from most discourse. Discussion of indispensability often proceeds as if the complete body of mathematical and scientific knowledge of the time were handed down from on high in some great volume. Had this been so, it would certainly seem baffling! But the actual history is nothing like this, mathematical and scientific understanding developed over millennia. To boot, much of contemporary mathematics would probably not be recognizable as mathematics at all by the ancients. It seems to me that the development of scientific theories is one of increasingly honing and making precise one's notions. It took great effort (by Galileo, Descartes, Newton, among others) to clarify the once poetic notions of motion, force, change, etc. The tools of calculus were no doubt conceived with the basic ideas of Newtonian mechanics in mind.
To my way of thinking, this kind of ever increasing "precisification" of one's concepts is necessary to the scientific enterprise. If the goal is understanding the physical world via some kind of explanatory theory (not a priori a mathematical one necessarily), then we are (most strongly) persuaded by one theory or another on the basis of its verifiably correct predictions. But the world is subtle and complicated [citation needed] and to make unambiguous verifiable predictions requires unambiguous conceptions in the first place. It seems to me that through this honing process by which one's concepts become more precise and rigorous, they *take on the character of mathematics*. This is one way that a subject or idea becomes a mathematical one (even if it was perhaps not so already). This phenomenon is not limited to physics. For ages, the notion of infinity was viewed as purely philosophical or poetic (not meant pejoratively). But in making various conceptions of the infinite precise (limits, infinite cardinalities, the ordinal numbers, etc.), they *became* mathematical notions which have in turn vastly sharpened the philosophical ideas.
Perhaps some combination of abstraction and rigour (and nothing else) is the defining character of mathematics. As I have suggested above, I believe that rigour is ultimately necessary for scientific reasoning and understanding. The role of abstraction is murkier maybe but, from experience, abstraction is often a key part of successful *explanations* (an enormous list of ultimately correct specific predictions would hardly count as an "explanation" of the physical phenomena). On this way of thinking, the mathematical flavour, on the one hand, of our most successful physical theories is as inevitable as their physical character, on the other; rigour and abstraction are part of the answer to the question "What is mathematics?" in the first place (maybe) and they are desirable features of our scientific theories.
I am certainly no expert and I would welcome scholarly references that engage with this line of thought. How is it that new mathematics emerges (or how do we come to understand what's already out there in the Platonic realm, if you prefer) and *might the answer to this question also hold the key to the role of mathematics in our scientific theories*?
Thanks very much for your comment. The question of how math emerges is central to many works in the philosophy of mathematics, and I think of this somewhat apart from the indispensibility argument. For example, my seminar students this week are reading Lakatos's Proofs and Refutations, whose central theme is just your question, since he describes a process by which mathematicians make progress, which departs dramatically from what he describes as the sterile formalist picture whereby mathematics proceeds by axioms and formal derivations of theorems.
I will definitely grab a copy of Lakatos! I of course did not mean to suggest that the literature is lacking in work on the emergence of mathematics, only that maybe there is a connection with the indispensability question: perhaps the desirable (if not necessary) features of a scientific theory are "automatically" leading to mathematical tools because these very features are characteristic of mathematics. (The most cogent examples being cases---like the development of calculus to explain motion/change or the set theoretic treatment of infinity to elucidate the infinite---where the notions with which the "new" mathematics is concerned were not previously considered to be a part of mathematics at all.) But this is far from a fully formed thought and maybe it does not withstand scrutiny or offer much insight after all.
There is an aspect of the indispensability question that has always struck me as conspicuously absent from most discourse. Discussion of indispensability often proceeds as if the complete body of mathematical and scientific knowledge of the time were handed down from on high in some great volume. Had this been so, it would certainly seem baffling! But the actual history is nothing like this, mathematical and scientific understanding developed over millennia. To boot, much of contemporary mathematics would probably not be recognizable as mathematics at all by the ancients. It seems to me that the development of scientific theories is one of increasingly honing and making precise one's notions. It took great effort (by Galileo, Descartes, Newton, among others) to clarify the once poetic notions of motion, force, change, etc. The tools of calculus were no doubt conceived with the basic ideas of Newtonian mechanics in mind.
To my way of thinking, this kind of ever increasing "precisification" of one's concepts is necessary to the scientific enterprise. If the goal is understanding the physical world via some kind of explanatory theory (not a priori a mathematical one necessarily), then we are (most strongly) persuaded by one theory or another on the basis of its verifiably correct predictions. But the world is subtle and complicated [citation needed] and to make unambiguous verifiable predictions requires unambiguous conceptions in the first place. It seems to me that through this honing process by which one's concepts become more precise and rigorous, they *take on the character of mathematics*. This is one way that a subject or idea becomes a mathematical one (even if it was perhaps not so already). This phenomenon is not limited to physics. For ages, the notion of infinity was viewed as purely philosophical or poetic (not meant pejoratively). But in making various conceptions of the infinite precise (limits, infinite cardinalities, the ordinal numbers, etc.), they *became* mathematical notions which have in turn vastly sharpened the philosophical ideas.
Perhaps some combination of abstraction and rigour (and nothing else) is the defining character of mathematics. As I have suggested above, I believe that rigour is ultimately necessary for scientific reasoning and understanding. The role of abstraction is murkier maybe but, from experience, abstraction is often a key part of successful *explanations* (an enormous list of ultimately correct specific predictions would hardly count as an "explanation" of the physical phenomena). On this way of thinking, the mathematical flavour, on the one hand, of our most successful physical theories is as inevitable as their physical character, on the other; rigour and abstraction are part of the answer to the question "What is mathematics?" in the first place (maybe) and they are desirable features of our scientific theories.
I am certainly no expert and I would welcome scholarly references that engage with this line of thought. How is it that new mathematics emerges (or how do we come to understand what's already out there in the Platonic realm, if you prefer) and *might the answer to this question also hold the key to the role of mathematics in our scientific theories*?
Thanks very much for your comment. The question of how math emerges is central to many works in the philosophy of mathematics, and I think of this somewhat apart from the indispensibility argument. For example, my seminar students this week are reading Lakatos's Proofs and Refutations, whose central theme is just your question, since he describes a process by which mathematicians make progress, which departs dramatically from what he describes as the sterile formalist picture whereby mathematics proceeds by axioms and formal derivations of theorems.
I will definitely grab a copy of Lakatos! I of course did not mean to suggest that the literature is lacking in work on the emergence of mathematics, only that maybe there is a connection with the indispensability question: perhaps the desirable (if not necessary) features of a scientific theory are "automatically" leading to mathematical tools because these very features are characteristic of mathematics. (The most cogent examples being cases---like the development of calculus to explain motion/change or the set theoretic treatment of infinity to elucidate the infinite---where the notions with which the "new" mathematics is concerned were not previously considered to be a part of mathematics at all.) But this is far from a fully formed thought and maybe it does not withstand scrutiny or offer much insight after all.