As for everything else, so for a mathematical theory: beauty can be perceived but not explained. Arthur Cayley
How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality? Max Wilhelm Dehn
One of the most intriguing features of mathematics is its applicability to empirical science. Every branch of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. It's not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these theories is also varied. Not only does mathematics help with empirical predictions, it allows elegant and economical statement of many theories. Indeed, so important is the language of mathematics to science, that it is hard to imagine how theories such as quantum mechanics and general relativity could even be stated without employing a substantial amount of mathematics.
From the rather remarkable but seemingly uncontroversial fact that mathematics is indispensable to science, some philosophers have drawn serious metaphysical conclusions. In particular, Quine and Putnam have argued that the indispensability of mathematics to empirical science gives us good reason to believe in the existence of mathematical entities. According to this line of argument, reference to (or quantification over) mathematical entities such as sets, numbers, functions and such is indispensable to our best scientific theories, and so we ought to be committed to the existence of these mathematical entities. To do otherwise is to be guilty of what Putnam has called "intellectual dishonesty". Moreover, mathematical entities are seen to be on an epistemic par with the other theoretical entities of science, since belief in the existence of the former is justified by the same evidence that confirms the theory as a whole (and hence belief in the latter). This argument is known as the Quine-Putnam indispensability argument for mathematical realism.
Mark Colyvan, Stanford Encyclopedia of Philosophy
Number is the within of all things. Pythagoras
Numbers are the highest degree of knowledge. It is knowledge itself. Plato
As a result of Cantor's developments, one could divide the mathematical community into three sorts. There were the finitists, typified by the attitudes of Aristotle or Gauss, who would only speak of potential infinities, not of actual infinities. Then there were the intuitionists like Kronecker and Brouwer who denied that there was any meaningful content to the notion of quantities that are anything but finite. Infinities are just potentialities that can never be actually realised. To manipulate them and include them within the realm of mathematics would be like letting wolves into the sheepfold. Then there were the transfinitists like Cantor himself, who ascribe the same degree of reality to actual completed infinities as they did to finite quantities. In between, there existed a breed of manipulative transfinitists, typified by Hilbert, who felt no compunction or need to ascribe any ontological status to infinities but admitted them as useful ingredients of mathematical formalism whose presence was useful in simplifying and unifying other mathematical theories. "No one," he predicted, "though he speak with the tongue of angels, will keep people from using the principle of the excluded middle."
From: Pi in the Sky, by John Barrow. Oxford University Press, 1992. p. 216
Poets do not go mad, but chess players do. Mathematicians go mad, and cashiers, but creative artists very seldom. I am not, as will be seen, in any sense attacking logic: I only say that this danger does lie in logic, not in imagination. -- G. K. Chesterton
In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In their unhappiness over the events of this world, some immerse themselves in a kind of self-sufficiency in mathematics. (Some have engaged in it for this reason alone.)
Stansilaw Ulam, Adventures of a Mathematician
Ordinals, Cardinals, Representation of Numbers in Language.
English: one/first ; two/second ; three/third ; four/fourth
French: un/premier ; deux/second or deuxième ; trois/troisième ; quatre/quatrième
German: ein/erste ; zwei/ander or zweite ; drei/dritter ; vier/vierte
Italian: uno/primo ; due/secondo ; tre/terzo ; quattro/quarto
In each of these four languages the words for 'one' and 'first' are quite distinct in form and emphasize the distinction between solitariness (one) and priority (being first). In Italian and the more old-fashioned German and French usage of ander and second, there is also a clear difference between the words used for 'two' and 'second', just as there is in English. This reflects the Latin root sense in English, French, and Italian of being second, this is, coming next in line, and this does not necessarily have an immediate association with two quantities. But when we get to three and beyond, there is a clear and simple relationship between the cardinal and ordinal words. Presumably this indicates that the dual aspect of number was appreciated by the time the concepts of 'threeness' and 'fourness' had emerged linguistically, following a period when only words describing 'oneness' and 'twoness' existed with greater quantities described by joining those words together as we described above.
In all the known languages of Indo-European origin, numbers larger than four are never treated as adjectives, changing their form according to the thing they are describing. But, numbers up to and including four are: we say they are 'inflected'. [...] a rather antiquated structure that barely survives in the modern forms of many Indo-European languages. For example, in French we find two words un and une corresponding to the English 'one' and they are used according to the gender of what is being counted. An analogous feature of language that certainly survives in English is the way in which different adjectives are associated with the same quantities of different things. We speak of a pair of shoes, a brace of pheasants, a yoke of oxen, or a couple of people, but we would never speak of a brace of chickens or a couple of shoes. [...]
We have seen that the distinction between cardinal and ordinal aspects of number and the use of inflected adjectives is clear up to the number four but conflated beyond that. [Footnote: In Finnish there are still two kinds of plural, as in classical Greek, Biblical Hebrew and Arabic: one for two things and another for more than two. Also interesting in this respect is the fact that there is no connection between the words for '2' and '½' in the Romance and Slavic languages (nor in Hungarian which is not an Indo-European language) but in all the European languages the words for '3' and '1/3', '4' and '1/4' and so on, are closely related, just as they are in English. This may indicate that the concept of a fraction, or the relation between a number and the concept of a ratio, only emerged after counting beyond 'two'.]
[...]
A curious speculation arises [...] to give special status to the number 8 - the total number of fingers excluding the thumbs - that many known languages originally possessed a base-8 system (which they later replaced by something better), because the word for the number 'nine' appears closely related to the word for 'new' suggesting that nine was a new number added to a traditional system. There are about twenty examples of this link, including Sanskrit, Persian, and the more familiar Latin, where we can see novus = 'new' and novem = 'nine'.
From: Pi in the Sky, by John Barrow. Oxford University Press, 1992. p. 37-38
