Reuben Hersh - The Mathematical Experience (1981)

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He rests his faith on rigorous proof; he believes that the difference between a correct proof and an incorrect one is an unmistakable and decisive difference. He can think of no condemnation more damning than to say of a student, "He doesn't even know what a proof is."

Reuben Hersh
The sciences of quantity and of space in their simpler forms are known as arithmetic and geometry.

Reuben Hersh
Some order begins to emerge from this chaos when the primes are considered not in their individuality but in the aggregate; one considers the social statistics of the primes and not the eccentricities of the individuals.

Reuben Hersh
Proof serves many purposes simultaneously.... Proof is respectability. Proof is the seal of authority. Proof, in its best instance, increases understanding by revealing the heart of the matter. Proof suggests new mathematics... Proof is mathematical power, the electric voltage of the subject which vitalizes the static assertions of the theorems.

Reuben Hersh
Proof is for cosmetic purposes and also to reduce somewhat the edge of insecurity on which one always lives.

Reuben Hersh
In the realm of ideas, of mental objects, those ideas whose properties are reproducible are called mathematical objects, and the study of mental objects with reproducible properties is called mathematics.

Reuben Hersh
If the number of theorems is larger than one can possibly survey, who can be trusted to judge what is "important?" One cannot have survival of the fittest if there is no interaction.

Reuben Hersh
It seems certain that there is a limit to the amount of living mathematics that humanity can sustain at any time. As new mathematical specialties arise, old ones will have to be neglected.

Reuben Hersh