The development of mathematics towards greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules.
Either mathematics is too big for the human mind or the human mind is more than a machine.
The piano's world encompasses glass-nerved virtuos and stomping barrel-housers in fedoras; it is a world of pasture and storm, of perfumed smoke, of liquid mathematics.
With the computer and programming languages, mathematics has newly-acquired tools, and its notation should be reviewed in the light of them. The computer may, in effect, be used as a patient, precise, and knowledgeable "native speaker" of mathematical notation.
The practice of first developing a clear and precise definition of a process without regard for efficiency, and then using it as a guide and a test in exploring equivalent processes possessing other characteristics, such as greater efficiency, is very common in mathematics. It is a very fruitful practice which should not be blighted by premature emphasis on efficiency in computer execution.
It is almost as hard to define mathematics as it is to define economics, and one is tempted to fall back on the famous old definition attributed to Jacob Viner, "Economics is what economists do," and say that mathematics is what mathematicians do. A large part of mathematics deals with the formal relations of quantities or numbers.
Mathematics brought rigor to Economics. Unfortunately, it also brought mortis.
The completion of a rigorous course in mathematics - it is not even necessary that the student does well in such a course - appears to be an excellent means of sharpening the mind and developing mental skills that are of general benefit.
Given the brief - and generally misleading - exposure most people have to mathematics at school, raising the public awareness of mathematics will always be an uphill battle.
The increased abstraction in mathematics that took place during the early part of this century was paralleled by a similar trend in the arts. In both cases, the increased level of abstraction demands greater effort on the part of anyone who wants to understand the work.
I firmly believe that mathematics does not exist outside of humans. It is something we, as a species, invent.
For all the time schools devote to the teaching of mathematics, very little (if any) is spent trying to convey just what the subject is about. Instead, the focus is on learning and applying various procedures to solve math problems. That's a bit like explaining soccer by saying it is executing a series of maneuvers to get the ball into the goal. Both accurately describe various key features, but they miss the what and the why of the big picture.
Though the structures and patterns of mathematics reflect the structure of, and resonate in, the human mind every bit as much as do the structures and patterns of music, human beings have developed no mathematical equivalent to a pair of ears. Mathematics can only be "seen" with the "eyes of the mind". It is as if we had no sense of hearing, so that only someone able to sight read music would be able to appreciate its patterns and harmonies.
In fact, the answer to the question "What is mathematics?" has changed several times during the course of history... It was only in the last twenty years or so that a definition of mathematics emerged on which most mathematicians agree: mathematics is the science of patterns.
A PhD in Mathematics is three years of guessing it wrong, plus one week of getting it right and writing a dissertation.
Just as music comes alive in the performance of it, the same is true of mathematics. The symbols on the page have no more to do with mathematics than the notes on a page of music. They simply represent the experience.
There are a lot of very strong connections with music and mathematics. They both can work in patterns and sequences and repetitions.
Music is mathematics, the mathematics of listening, mathematics for the ears.
Mathematics is, as it were, a sensuous logic, and relates to philosophy as do the arts, music, and plastic art to poetry.
— Karl Wilhelm Friedrich Schlegel
I knew my purpose well and clear: to show how Nature behaves without cluttering its beauty with abtruse mathematics.
Mathematics is one of the surest ways for a man to feel the power of thought and the magic of the spirit. Mathematics is one of the eternal truths and, as such, raises the spirit to the same level on which we feel the presence of God.
— Julio Cesar de Mello e Souza
'Without the help of mathematics,' the wise man continued, 'the art could not advance and all the sciences would perish.'
— Julio Cesar de Mello e Souza
Cooking is just as creative and imaginative an activity as drawing, or wood carving, or music. And cooking draws upon your every talent-science, mathematics, energy, history, experience-and the more experience you have, the less likely are your experiments to end in drivel and disaster. The more you know, the more you can create.
Woe be to him who tries to isolate one department of knowledge from the rest. All science is one: language, literature and history, physics, mathematics and philosophy; subjects which seem the most remote from one another are in reality connected, or rather they all form a single system.
If I had inherited a fortune I should probably not have cast my lot with mathematics.
The general mental qualification necessary for scientific advancement is that which is usually denominated "common sense," though added to this, imagination, induction, and trained logic, either of common language or of mathematics, are important adjuncts.
Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.
This is an extremely ambitious book. In addition to science and mathematics, Byers brings to bear insights from literature, philosophy, religion, history, anthropology, medicine, and psychology. The Blind Spot breaks new ground, and represents a major step forward in the philosophy of science. The book is also a page-turner, which is rare for this topic.
Where mathematics and spirit join, where proof of the existence of mystery-salvific mystery-shimmers just below the surfaces of human perception, experience and the linguistic veil itself, Killarney Clary's new book-her best to date-dwells, plumbs, persuades and thrills.
If mathematics is to be understood widely, we need to emphasise its elegance and its applications. Sometimes it seems that universities want to emphasise how difficult it is!
Examples ... which might be multiplied ad libitum, show how difficult it often is for an experimenter to interpret his results without the aid of mathematics.
It is exceptional that one should be able to acquire the understanding of a process without having previously acquired a deep familiarity with running it, with using it, before one has assimilated it in an instinctive and empirical way... Thus any discussion of the nature of intellectual effort in any field is difficult, unless it presupposes an easy, routine familiarity with that field. In mathematics this limitation becomes very severe.
By and large it is uniformly true that in mathematics there is a time lapse between a mathematical discovery and the moment it becomes useful; and that this lapse can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful.
The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.
When we talk mathematics, we may be discussing a secondary language built on the primary language of the nervous system.
The total subject of mathematics is clearly too broad for any of us. I do not think that any mathematician since Gauss has covered it uniformly and fully; even Hilbert did not and all of us are of considerably lesser width quite apart from the question of depth than Hilbert.
Young man, in mathematics you don't understand things. You just get used to them.
If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.
The most vitally characteristic fact about mathematics is, in my opinion, its quite peculiar relationship to the natural sciences, or more generally, to any science which interprets experience on a higher than purely descriptive level.
Neumann, to a physicist seeking help with a difficult problem: Simple. This can be solved by using the method of characteristics. Physicist: I'm afraid I don't understand the method of characteristics. Neumann: In mathematics you don't understand things. You just get used to them.