How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably appropriate to the objects of reality?

Mathematics in itself, as I say, is independent of experience. It begins with the free choice of symbols, to which are freely assigned properties, and it then proceeds to deduce the necessary rational implications of those properties.

Underpinning everything... are the laws of physics. These remarkably ingenious laws are able to permit matter to self-organize to the point where consciousness emerges in the cosmos—mind from matter—and the most striking product of the human mind is mathematics. This is the baffling thing. Mathematics is... produced by the human mind. Yet if we ask where mathematics works best, it is in areas like particle physics and astrophysics, areas of fundamental science that are very, very far removed from everyday affairs. ...at the opposite end of spectrum of complexity from the human brain. ...a product of the most complex system we know in nature, the human brain, finds a consonance with the underlying, simplest and most fundamental level, the basic building blocks that make up the world.

If mathematical objects exist independently of human thought what else are they independent of? The universe, I suppose.

Mathematics is primarily a tool for human thought.

For scholars and laymen alike it is not philosophy but active experience in mathematics itself that can alone answer the question: What is mathematics?

Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.

4. Human mathematics is the cultural product of a community of rational beings.

The odd belief prevails in our culture that a thing or experience is not real if we cannot make it mathematical, and somehow it must be real if we can reduce it to numbers. But this means making an abstraction out of it - mathematics is the abstract par excellence, which is indeed its glory and the reason for its great usefulness.

The Mathematics which effectually exercises, not vainly deludes or vexatiously torments studious Minds with obscure Subtilties, perplexed Difficulties, or contentious Disquisitions; which overcomes without Opposition, triumphs without Pomp, compels without Force, and rules absolutely without Loss of Liberty; which does not privately overreach a weak Faith, but openly assaults an armed Reason, obtains a total Victory, and puts on inevitable Chains; whose Words are so many Oracles, and Works as many Miracles; which blabs out nothing rashly, nor designs anything from the Purpose, but plainly demonstrates and readily performs all Things within its Verge; which obtrudes no false Shadow of Science, but the very Science itself, the Mind firmly adheres to it, as soon as possessed of it, and can never after desert it of its own Accord, or be deprived of it by any Force of others: Lastly the Mathematics, which depend upon Principles clear to the Mind, and agreeable to Experience; which draws certain Conclusions, instructs by profitable Rules, unfolds pleasant Questions; and produces wonderful Effects; which is the fruitful Parent of, I had almost said all, Arts, the 47 unshaken Foundation of Sciences, and the plentiful Fountain of Advantage to human Affairs.

Although mathematics originates in the human mind, its remarkable effectiveness in explaining the world does not extend to the mind itself. Psychology has proved unusually resistant to the mathematization that works so well in physics.

Mathematics, like music and poetry, is a creation of the mind; … the primary task of the mathematician, like that of any other artist, is to extend man's mental horizon by representation and interpretation.

6. Mathematics is applicable because it truly describes the fundamental structure of reality.

As a science mathematics has been adapted to the description of natural phenomena, and the great practitioners in this field... have never concerned themselves with the logical foundations of mathematics, but have boldly taken a pragmatic view of mathematics as an intellectual machine which works successfully. Description has been verified by further observation, still more strikingly be prediction, and sometimes, more ominously, by control of natural forces. Happily, unresolved problems... still remain as challenges.