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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. | 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 | + | 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. |
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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. | ||
Revision as of 07:51, 28 September 2009
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.
