"The present work has two main objects. One of these, the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles... and will be established by strict symbolic reasoning... The demonstration of this thesis has, if I am not mistaken, all the certainty and precision of which mathematical demonstrations are capable. As the thesis is very recent among mathematicians, and is almost universally denied by philosophers, I have undertaken... to defend... against such adverse theories as appeared most widely held or most difficult to disprove. I have also endeavoured to present, in language as untechnical as possible, the more important stages in the deductions by which the thesis is established. The other object of this work... is the explanation of the fundamental concepts which mathematics accepts as indefinable. This is a purely philosophical task, and I cannot flatter myself that I have done more than indicate a vast field of inquiry, and give a sample of the methods by which the inquiry may be conducted. The discussion of indefinables—which forms the chief part of philosophical logic—is the endeavour to see clearly... the entities concerned, in order that the mind may have that kind of acquaintance with them which it has with redness or the taste of a pineapple. Where, as in the present case, the indefinables are obtained primarily as the necessary residue in a process of analysis, it is often easier to know that there must be such entities than actually to perceive them..."
Foundations of mathematics are the logical and mathematical frameworks that allow the development of mathematics without generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particular. This may also include the philosophical study of the relation of this framework with reality.
"For the mathematician the important consideration is that the foundations of mathematics and a great portion of its content are Greek. The Greeks laid down the first principles, invented the methods ab initio, and fixed the terminology. Mathematics in short is a Greek science, whatever new developments modern analysis has brought or may bring."
"In the summer of 1914 I attended Freges course, Logik in der Mathematik. Here he examined critically some of the customary conceptions and formulations in mathematics. He deplored the fact that mathematicians did not even seem to aim at the construction of a unified, well-founded system of mathematics, and therefore showed a lack of interest in foundations. He pointed out a certain looseness in the customary formulation of axioms, definitions, and proofs, even in the works of the more prominent mathematicians. As an example he quoted Weyerstrasss definition: "A number is a series of things of the same kind"... On this he commented with an impish smile: "According to this definition, a railroad train is also a number; this number may then travel from Berlin, pass through Jena... He criticized in particular the lack of attention to certain fundamental distinctions, e.g., ...between the symbol and the symbolized, ...between a logical concept and a mental image or act, and that between a function and the value of a function. Unfortunately, his admonitions go unheeded even today."
"An article by Henri Poincaré entitled The Nature of Mathematical Reasoning... appeared in 1894 as the first of a series of investigations into the foundations of the exact sciences. It was a signal for a throng of other mathematicians to inaugurate a movement for the revision of the classical concepts, a movement which culminated in the nearly complete absorption of logic into the body of mathematics."
"I have also in my leisure hours frequently reflected upon another problem, now of nearly forty years standing. I refer to the foundations of geometry. I do not know whether I have ever mentioned to you my views on this matter. My meditations here also have taken more definite shape, and my conviction that we cannot thoroughly demonstrate geometry a priori is, if possible, more strongly confirmed than ever. But it will take a long time for me to bring myself to the point of working out and making public my very extensive investigations on this subject, and possibly this will not be done during my life, inasmuch as I stand in dread of the clamors of the Bœotians, which would be certain to arise, if I should ever give full expression to my views. It is curious that in addition to the celebrated flaw in Euclids Geometry, which mathematicians have hitherto endeavored in vain to patch and never will succeed, there is still another blotch in its fabric to which, so far as I know, attention has never yet been called and which it will by no means be easy, if at all possible, to remove. This is the definition of a plane as a surface in which a straight line joining any two points lies wholly in that plane. This definition contains more than is requisite to the determination of a surface, and tacitly involves a theorem which is in need of prior proof."
"The ease with which you have assimilated my notions of geometry has been a source of genuine delight to me, especially as so few possess a natural bent for them. I am profoundly convinced that the theory of space occupies an entirely different position with regard to our knowledge a priori from that of the theory of numbers (Grössenlehre); that perfect conviction of the necessity and therefore the absolute truth which is characteristic of the latter is totally wanting to our knowledge of the former. We must confess in all humility that a number is solely a product of our mind. Space, on the other hand, possesses also a reality outside of our mind, the laws of which we cannot fully prescribe a priori."
"Much has been written on the history of calculus... However, historians tend to harp on the question of logical justification and to spend a disproportionate amount of time on the way it was handled in the nineteenth century. This not only obscures the boldness and vigor of early calculus, but it is overly dogmatic about the way in which calculus should be justified. ...the sheer diversity of foundations for calculus suggests that we have not yet got to the bottom of it."