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We might tentatively define mathematics as a particular sort of logical system, often but not necessarily concerned with number and shape, that seeks to derive high level generalizations from simple axioms and definitions. The power of the discipline can be found both in its generality and its rigor: a theorem which has been proved usually says something about an infinite number of objects, and says it with unrivaled specificity and clarity. To utilize the computational strength of mathematics, we have to distill a phenomenon into its most essential elements, and at times transform a straightforward concrete event into an object of high abstraction. But having satisfied those requirements, we are given a framework within which profound mental feats can be accomplished. In this sense, mathematics is a tool that, properly wielded, bootstraps the human intellect into doing things beyond its ordinary powers. Impressive as human minds are, they can only juggle so many concepts simultaneously, can only perform so many operations per second, and can only maintain focus for so long. Mathematics allows us to offload difficult computation into the machinery of previously established theorems, whose if-then statements churn out desired information from previously known, less useful values. | We might tentatively define mathematics as a particular sort of logical system, often but not necessarily concerned with number and shape, that seeks to derive high level generalizations from simple axioms and definitions. The power of the discipline can be found both in its generality and its rigor: a theorem which has been proved usually says something about an infinite number of objects, and says it with unrivaled specificity and clarity. To utilize the computational strength of mathematics, we have to distill a phenomenon into its most essential elements, and at times transform a straightforward concrete event into an object of high abstraction. But having satisfied those requirements, we are given a framework within which profound mental feats can be accomplished. In this sense, mathematics is a tool that, properly wielded, bootstraps the human intellect into doing things beyond its ordinary powers. Impressive as human minds are, they can only juggle so many concepts simultaneously, can only perform so many operations per second, and can only maintain focus for so long. Mathematics allows us to offload difficult computation into the machinery of previously established theorems, whose if-then statements churn out desired information from previously known, less useful values. | ||
| − | When a student is doing calculus, she can use a formula to integrate functions without referencing any of the technical set theoretic underpinnings of real numbers, or even being concerned with knowing how the integral formula is derived. A good student who really wants to understand the material should have a strong familiarity with how integration formulas are proved, but it is not required to obtain the correct answer, and herein lies the clever move that mathematics performs. Once a certain theorem is proved, it is proved forever, and the laborious calculation required to reach the theorem can in some sense be forgotten. It is as though a mathematician uses a ladder to reach a higher point, and then kicks the ladder out from underneath him. The effect of this phenomenon is that high level conceptual work can be done without going through the thousands of individual calculations and proofs that are implicit in even minor operations and steps. | + | When a student is doing calculus, she can use a formula to integrate functions without referencing any of the technical set theoretic underpinnings of real numbers, or even being concerned with knowing how the integral formula is derived. A good student who really wants to understand the material should have a strong familiarity with how integration formulas are proved, but it is not required to obtain the correct answer, and herein lies the clever move that mathematics performs. Once a certain theorem is proved, it is proved forever, and the laborious calculation required to reach the theorem can in some sense be forgotten. It is as though a mathematician uses a ladder to reach a higher point, and then kicks the ladder out from underneath him. The effect of this phenomenon is that high level conceptual work can be done without going through the thousands of individual calculations and proofs that are implicit in even minor operations and steps. The mental lifting that is accomplished in using something like Stoke's theorem is, when viewed with extreme magnification, stupendous. It would be premature to here decide the limits of human intellectual ability, but I think I am safe in saying that no person who has ever lived could reason ''directly'' from set theory to, say, research level topology, without first arduously learning the intermediate theorems. Through the mathematical paradigm of proof, then, we transcend our ordinary limitations and gain access to higher level conceptual manipulation. |
| − | + | Our question concerning why mathematics accurately captures the essence of physical phenomena might be informed by this argument. Adopting an engineering standpoint, we can reasonably say that human minds were shaped by evolution to solve a certain set of problems faced in our ancestral environments. We successfully developed perceptual and analytical abilities for handling issues in social group dynamics, food finding and self preservation generally. But the talents with which we have been endowed were, over the course of history, exapted to purposes never anticipated by natural selection. | |
| − | + | to the extent that our initial axioms reflect the fundamental properties of nature, higher level mathematics will correspondingly mimic the emergent properties of the physical world. | |
It seems to be the opinion of (tendency toward simplicity; models of the universe at the fundamental level have less freedom due to their simplicity; higher probability of getting the model right.) | It seems to be the opinion of (tendency toward simplicity; models of the universe at the fundamental level have less freedom due to their simplicity; higher probability of getting the model right.) | ||
Revision as of 12:04, 28 September 2009
David Knott (title later)
One of the most peculiar characteristics of mathematics is its seemingly limitless ability to accurately account for real world phenomena. The success of the discipline in providing a rigorous structure on which principles of physics and chemistry can be scaffolded is perhaps most strongly demonstrated by the justification frequently given for studying theoretical math: although some given piece of research may have no connection to the real world, there is a high probability that one will be found at a future time. The confidence that researchers have in the usefulness of mathematics, even before an actual use has been found, speaks volumes about the strong parallels between mathematical principles and the underlying architecture of reality.
This is a concept we are accustomed to. Few and unfortunate are the students who walk into a physics course not expecting a heavily mathematical treatment of the subject matter. And virtually no one believes that the much sought after physical "theory of everything" will be properly described without reference to math. But there is no a priori reason for this to be the case. Why does mathematics so elegantly reveal the nature of our world? This is a question that I cannot dream of answering, but it is sometimes useful to ineffectually bash one's head against a problem in the hope that something insightful pops out. I will begin the head bashing, then, by talking a little about what mathematics is.
We might tentatively define mathematics as a particular sort of logical system, often but not necessarily concerned with number and shape, that seeks to derive high level generalizations from simple axioms and definitions. The power of the discipline can be found both in its generality and its rigor: a theorem which has been proved usually says something about an infinite number of objects, and says it with unrivaled specificity and clarity. To utilize the computational strength of mathematics, we have to distill a phenomenon into its most essential elements, and at times transform a straightforward concrete event into an object of high abstraction. But having satisfied those requirements, we are given a framework within which profound mental feats can be accomplished. In this sense, mathematics is a tool that, properly wielded, bootstraps the human intellect into doing things beyond its ordinary powers. Impressive as human minds are, they can only juggle so many concepts simultaneously, can only perform so many operations per second, and can only maintain focus for so long. Mathematics allows us to offload difficult computation into the machinery of previously established theorems, whose if-then statements churn out desired information from previously known, less useful values.
When a student is doing calculus, she can use a formula to integrate functions without referencing any of the technical set theoretic underpinnings of real numbers, or even being concerned with knowing how the integral formula is derived. A good student who really wants to understand the material should have a strong familiarity with how integration formulas are proved, but it is not required to obtain the correct answer, and herein lies the clever move that mathematics performs. Once a certain theorem is proved, it is proved forever, and the laborious calculation required to reach the theorem can in some sense be forgotten. It is as though a mathematician uses a ladder to reach a higher point, and then kicks the ladder out from underneath him. The effect of this phenomenon is that high level conceptual work can be done without going through the thousands of individual calculations and proofs that are implicit in even minor operations and steps. The mental lifting that is accomplished in using something like Stoke's theorem is, when viewed with extreme magnification, stupendous. It would be premature to here decide the limits of human intellectual ability, but I think I am safe in saying that no person who has ever lived could reason directly from set theory to, say, research level topology, without first arduously learning the intermediate theorems. Through the mathematical paradigm of proof, then, we transcend our ordinary limitations and gain access to higher level conceptual manipulation.
Our question concerning why mathematics accurately captures the essence of physical phenomena might be informed by this argument. Adopting an engineering standpoint, we can reasonably say that human minds were shaped by evolution to solve a certain set of problems faced in our ancestral environments. We successfully developed perceptual and analytical abilities for handling issues in social group dynamics, food finding and self preservation generally. But the talents with which we have been endowed were, over the course of history, exapted to purposes never anticipated by natural selection.
to the extent that our initial axioms reflect the fundamental properties of nature, higher level mathematics will correspondingly mimic the emergent properties of the physical world.
It seems to be the opinion of (tendency toward simplicity; models of the universe at the fundamental level have less freedom due to their simplicity; higher probability of getting the model right.)
