Concomitant with these general questions are many more specific ones. Are the integer 28, the real number 28.0, the complex number 28.0 + 0i, the 1 x 1 matrix ; and the constant function f(x)=28 the same entity or different entities? Different programming languages have different answers. Is ``the integer 28'' a single entity or a collection of similar entities; the signed integer, the whole number, the ordinal, the cardinal, and so on? Did Euler mean the same thing that we do when he wrote an integral sign? For that matter, do a contemporary measure theorist, a PDE expert, a numerical analyst, and a statistician all mean the same thing when they use an integral sign?
Such questions have been debated among philosophers and mathematicians for at least two and a half millennia. But, though the questions are eternal, the answers may not be. The standpoint from which we view these issues is significantly different from Hilbert and Poincaré, to say nothing of Newton and Leibnitz, Plato and Pythagoras, reflecting the many changes the last century has brought. Mathematics itself has changed tremendously: vast new areas, new techniques, new modes of thought have opened up, while other areas have been largely abandoned. The applications and misapplications of mathematics to the sciences, engineering, the arts and humanities, and society have exploded. The electronic computer has arrived and has transformed the landscape. Computer technology offers a whole collection of new opportunities, new techniques, and new challenges for mathematical resesarch; it also brings along its own distinctive viewpoint on mathematics.
The past century has also seen enormous growth in our understanding of mathematics and mathematical concepts as a cognitive and cultural phenomenon. A great deal is now known about the psychology and even the neuroscience of basic mathematical ability; about mathematical concepts in other cultures; about mathematical reasoning in children, in pre-verbal infants, and in animals.
Moreover the larger intellectual environment has altered, and with it, our views of truth and knowledge generally. Works such as Kuhn's analysis of scientific progress and Foucault's analysis of the social aspects of knowledge have become part of the general intellectual currency. One can decide to reject them, but one cannot ignore them.