Mathematics and Science

The following sections are included:

• Preface

• Contents

The following sections are included:

• On the “First Principles” of Physics and Mathematics

• Progress and Limitations

• Physics and Mathematics in the Theory of Elementary Particles

• Bibliography

When Sir James Jeans proclaimed that ‘God is a mathematician’, he articulated the physicist's conviction that the fundamental laws of the physical universe are expressible as simple mathematical statements. Why this should be so has fascinated generations of scientists. I believe, however, that the efficacy of mathematics in describing the physical world is only part of a larger mystery, which is why the universe is knowable at all…

The following sections are included:

• Remarks on Sociology

• Scientific Revolutions

• The Distinction between Mathematical Models and Quantification

• Reasons for the Relative Absence of Mathematics in Sociology
  • Spurious Reasons
  • Non-Spurious Reasons

• Chances for Change
  • Specialty Fields
  • Structural Mathematics

• Conclusion

• References

The following sections are included:

• What is Mathematics?

• What is Science?

• Fundamental Roles of Mathematics in Science

• Misuses of Mathematics

Here is an old conundrum, many times resurrected: why do mathematics and physics fit together so surprisingly well? There is a famous article by Eugene Wigner, or at least an article with a famous title: “The Unreasonable Effectiveness of Mathematics in Natural Sciences.” After all, pure mathematics, as we all know, is created by fanatics sitting at their desks or scribbling on their blackboards. These wild men go where they please, led only by some notion of “beauty”, “elegance”, or “depth”, which nobody can really explain. Wigner wrote, “It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind's capacity to divine them.”…

The following sections are included:

• The Nature of Mathematics

• Mathematics and Nature

• Scientific Realism

• Mathematical Structure in the Quantum World

• Summary

• References

The following sections are included:

• What is Mathematics?

• The Construction of Mathematics

• Mathematics from the Viewpoint of Systems

• A Description of the States of Materials

• Some Epistemological Problems

• A Mathematical Discussion of a Social Phenomenon

• Final Comments and Further Questions

• Acknowledgment

• References

That famous quotation from Eugene Wigner – on the unreasonable effectiveness of mathematics – came at a time when the standard attitude towards the nature of mathematics was primarily formal and hence misleading. Indeed, the view then dominant took mathematics to be strictly formal, partly because Bertrand Russell had opined that mathematics was not knowledge of the truth, but just correct deduction of conclusions from premises, true or not…

The following sections are included:

• Introduction

• Infinite Idealizations

• Empirical Bases for Number Systems
  • Extensive Measurement
  • One Empirical Base for Multiplication: Multiplicative Representations
  • A Second Empirical Base for Multiplication: Automorphisms
  • A Third Empirical Basis for Multiplication: Distributive Operations
  • What Else is Measurable?
  • Distribution of General Relational Structures in Conjoint Ones

• The Use of Non-Empirical Mathematics

• Conclusion

• References

The following sections are included:

• Introduction

• Definitions

• Four Propositions about Nature and Knowledge

• Key Questions on the Math-Science Relationship

• References

The following sections are included:

• Introduction

• The Role of Mathematics

• Mathematics as Language

• Mathematics beyond Language

• Mathematics and Music

• Mathematics and the Brain

• Mathematics and Archetypes

• Conclusion

• References

The properties of subatomic matter are currently understood in terms of what is called the “standard model”.¹ This is obtained by the juxtaposition of two theories, each successful within its own limited domain. One is quantum chromodynamics (QCD), the theory of the interactions of those quarks and gluons which are believed to be the constituents of nuclear matter. The second theory is the unified theory of Salam and Weinberg, which synthesized weak and electromagnetic interactions into a single scheme…

The following sections are included:

• Introduction

• Natural Systems and Formal Systems

• Entailment in Natural and Formal Systems

• Natural Law

• Modelling Relations

• Some Ramifications of Modelling Relations
  • Analogous Systems
  • Alternate Models

• A Postscript: The Effectiveness of Mathematics in Mathematics

• References

Mathematics, which has been called “unreasonably effective”, sometimes develops on its own, with problems and goals stemming from its own subject matter, and sometimes develops in tandem with other disciplines. In the latter case, mathematical structures may be articulated in response to the needs of other theories in natural science or may actually reveal directions in which the development of physical theories should proceed. No one doubts that such ventures – at least since the “scientific revolution” that began at the end of the Renaissance – have been successful. In fact, the spurt in the development of mathematics around this time is sometimes said to have made this revolution possible…

A general theme of the present volume is to be “why and how mathematics works to describe reality. These questions are perplexing enough even in the physical sciences where mathematics has been used with the greatest success. The employment of mathematics in the biological sciences is perhaps somewhat less pristine…

Reading Eugene Wigner's essay¹ on the role of mathematics in science, the physics teacher is struck by the sentence: “(Newton's law of planetary motion,) particularly since a second derivative appears in it, is simple only to the mathematician, not to common sense or to non-mathematically-minded freshmen.” The statement is undoubtedly true, but why? Why do freshmen seem to have a harder time with physics than with French, with political science, or, for that matter, with mathematics? What is so difficult about physics? Thinking about this question leads to three observations that bear on the more general problem set forth by Wigner…

The following sections are included:

• Introduction

• The Linear World View

• Chaos and the World View

• Further Ruminations

• References

There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. “How can you know that?” was his query…

When I was a kid, so to speak, I came across and read Eugene Wigner's article¹ on “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”. I was impressed. Thus, I was rather pleased when Ron Mickens asked me, this many years later, to write a similar article…

The following sections are included:

• Bibliography

• Subject Index