A Quote by Morris Kline
Though determinants and matrices received a great deal of attention in the nineteenth century and thousands of papers were written on these subjects, they do not constitute great innovations in mathematics.... Neither determinants nor matrices have influenced deeply the course of mathematics despite their utility as compact expressions and despite the suggestiveness of matrices as concrete groups for the discernment of general theorems of group theory.
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It is my experience that proofs involving matrices can be shortened by 50% if one throws the matrices out.
It was not until some weeks later that I realized there is no need to restrict oneself to 2 by 2 matrices. One could go on to 4 by 4 matrices, and the problem is then easily soluable. In retrospect, it seems strange that one can be so much held up over such an elementary point. The resulting wave equation for the electron turned out to be very successful. It led to correct values for the spin and the magnetic moment. This was quite unexpected. The work all followed from a study of pretty mathematics, without any thought being given to these physical properties of the electron.
Mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It's the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.
Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems-general and specific statements-can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences.
...the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly generality is, in essence, the same as a small and concrete special case.
The phenomenon of creativity, we know, is closely related to the ability to yoke together separate, and even seemingly incompatible, matrices.
I like to do matrices. One option per line, different facets for each column. Salary, location, happiness index, failure index, and all that.
Come, every frustum longs to be a cone, And every vector dreams of matrices. Hark to the gentle gradient of the breeze: It whispers of a more ergodic zone.
To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples.
Health inequalities and the social determinants of health are not a footnote to the determinants of health. They are the main issue.
As Enlightenment philosophers and scholars consciously adopted the methods of science to establish such abstract concepts as rights, liberty, and justice, successive generations have become schooled in thinking of these abstractions as applied to others in matrices-like mental rotations.
We [he and Halmos] share a philosophy about linear algebra: we think basis-free, we write basis-free , but when the chips are down we close the office door and compute with matrices like fury.
General Systems Theory is a name which has come into use to describe a level of theoretical model-building which lies somewhere between the highly generalized constructions of pure mathematics and the specific theories of the specialized disciplines. Mathematics attempts to organize highly general relationships into a coherent system, a system however which does not have any necessary connections with the "real" world around us. It studies all thinkable relationships abstracted from any concrete situation or body of empirical knowledge.
The liver, that great maroon snail: No wave of emotion sweeps it. Neither music nor mathematics gives it pause in its appointed tasks.
One may say that mathematics talks about the things which are of no concern to men. Mathematics has the inhuman quality of starlight - brilliant, sharp but cold ... thus we are clearest where knowledge matters least: in mathematics, especially number theory.
The heart of mathematics consists of concrete examples and concrete problems. Big general theories are usually afterthoughts based on small but profound insights; the insights themselves come from concrete special cases.
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