"Intuition isn't direct perception of something external. It's the effect in the mind/brain of manipulating concrete objects - at a later stage, of making marks on paper, and still later, manipulating mental images. This experience leaves a trace, an effect, in your mind/brain." (Reuben Hersh, "What Is Mathematics, Really?", 1998)
"A mathematical entity is a concept, a shared thought. Once you have acquired it, you have it available, for inspection or manipulation. If you understand it correctly (as a student, or as a professional) your ‘mental model’ of that entity, your personal representative of it, matches those of others who understand it correctly. (As is verified by giving the same answers to test questions.) The concept, the cultural entity, is nothing other than the collection of the mutually congruent personal representatives, the ‘mental models’, possessed by those participating in the mathematical culture." (Reuben Hersh, "Experiencing Mathematics: What Do We Do, when We Do Mathematics?", 2014)
"A coherent inclusive study of the nature of mathematics would contribute to our understanding of problem-solving in general. Solving problems is how progress is made in all of science and technology. The synthesizing energy to achieve such a result would be a worthy and inspiring task for philosophy." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)
"A mathematician possesses a mental model of the mathematical entity she works on. This internal mental model is accessible to her direct observation and manipulation. At the same time, it is socially and culturally controlled, to conform to the mathematics community's collective model of the entity in question. The mathematician observes a property of her own internal model of that mathematical entity. Then she must find a recipe, a set of instructions, that enables other competent, qualified mathematicians to observe the corresponding property of their corresponding mental model. That recipe is the proof. It establishes that property of the mathematical entity." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)
"Different models are both competitive and complementary. Their standing will depend on their benefits in practice. If philosophy of mathematics were seen as modeling rather than as taking positions, it might consider paying attention to mathematics research and mathematics teaching as testing grounds for its models." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)
"History of mathematics is done by mathematicians as well as historians. History models mathematics as a segment of the ongoing story of human culture. Mathematicians are likely to see the past through the eyes of the present, and ask, ‘Was it important? natural? deep? surprising? elegant?’ The historian sees mathematics as a thread in the ever-growing web of human life, intimately interwoven with finance and technology, with war and peace. Today's mathematics is the culmination of all that has happened before now, yet to future viewpoints it will seem like a brief, outmoded stage of the past." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)
"Logic sees mathematics as a collection of virtual inscriptions - declarative sentences that could in principle be written down. On the basis of that vision, it offers a model: formal deductions from formal axioms to formal conclusions--formalized mathematics. This vision itself is mathematical. Mathematical logic is a branch of mathematics, and whatever it's saying about mathematics, it is saying about itself--self-reference." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)
"[...] mathematics is too complex, varied and elaborate to be encompassed in any model. An all-inclusive model would be like the map in the famous story by Borges - perfect and inclusive because it was identical to the territory it was mapping." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)
"Mathematical modeling is the modern version of both applied mathematics and theoretical physics. In earlier times, one proposed not a model but a theory. By talking today of a model rather than a theory, one acknowledges that the way one studies the phenomenon is not unique; it could also be studied other ways. One's model need not claim to be unique or final. It merits consideration if it provides an insight that isn't better provided by some other model." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)
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