Philosophy And Mathematics Quotes & Sayings

When we are young, we spend much time and pains in filling our note-books with all definitions of Religion, Love, Poetry, Politics, Art, in the hope that, in the course of a few years, we shall have condensed into our encyclopaedia the net value of all the theories at which the world has yet arrived. But year after year our tables get no completeness, and at last we discover that our curve is a parabola, whose arcs will never meet. — Ralph Waldo Emerson

As John Adams famously wrote during the American Revolution, "I must study politics and war, that our sons may have liberty to study mathematics and philosophy. Our sons ought to study mathematics and philosophy, geography, natural history and naval architecture, navigation, commerce and agriculture in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry and porcelain." So maybe today they're writing apps rather than studying poetry, but that's an adjustment for the age. — Fareed Zakaria

India was the motherland of our race
and Sanskrit the mother of Europe's languages.
India was the mother of our philosophy,
of much of our mathematics, of the ideals embodied in
Christianity ... of self-government and democracy.
In many ways, Mother India is the mother of us all. — Will Durant

Our species is the only creative species, and it has only one creative instrument, the individual mind and spirit of man. Nothing was ever created by two men. There are no good collaborations, whether in music, in art, in poetry, in mathematics, in philosophy. Once the miracle of creation has taken place, the group can build and extend it, but the group never invents anything. The preciousness lies in the lonely mind of a man. — John Steinbeck

Everything we learn - economics, philosophy, biology, mathematics - has to be understood in light of the overarching reality of the character of God. That is why, in the Middle Ages, theology was called "the queen of the sciences" and philosophy "her handmaiden." Today the queen has been deposed from her throne and, in many cases, driven into exile, and a supplanter now reigns. We have replaced theology with religion. — R.C. Sproul

If your wish is to become really a man of science and not merely a petty experimentalist, I should advise you to apply to every branch of natural philosophy, including mathematics. — Mary Wollstonecraft Shelley

The physical method becomes a philosophy when it asserts there is no higher knowledge than the empirical knowledge of scientific phenomena. The mathematical method becomes a philosophy when it asserts that some higher knowledge is needed to explain scientific facts, and that higher knowledge is mathematics. — Fulton J. Sheen

The most ignorant person, at a reasonable charge, and with a little bodily labour, might write books in philosophy, poetry, politics, laws, mathematics, and theology, without the least assistance from genius or study. — Jonathan Swift

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. — Joseph Auslander

Religion is but a single brand of superstition (others include beliefs in astrology, paranormal phenomena, homeopathy, and spiritual healing), but it is the most widespread and harmful form of superstition. And science is but one form of rationality (philosophy and mathematics are others), but it is a highly developed form, and the only one capable of describing and understanding reality. — Jerry A. Coyne

My parents owned a pharmacy in Budapest, which gave us a comfortable living. As I was their only child, they wanted me to become a pharmacist. But my own preference would have been to study philosophy and mathematics. — John Harsanyi

Our craving for generality has [as one] source ... our preoccupation with the method of science. I mean the method of reducing the explanation of natural phenomena to the smallest possible number of primitive natural laws; and, in mathematics, of unifying the treatment of different topics by using a generalization. Philosophers constantly see the method of science before their eyes, and are irresistibly tempted to ask and answer in the way science does. This tendency is the real source of metaphysics, and leads the philosopher into complete darkness. I want to say here that it can never be our job to reduce anything to anything, or to explain anything. Philosophy really is purely descriptive. — Ludwig Wittgenstein

We find sects and parties in most branches of science; and disputes which are carried on from age to age, without being brought to an issue. Sophistry has been more effectually excluded from mathematics and natural philosophy than from other sciences. In mathematics it had no place from the beginning; mathematicians having had the wisdom to define accurately the terms they use, and to lay down, as axioms, the first principles on which their reasoning is grounded. Accordingly, we find no parties among mathematicians, and hardly any disputes. — Thomas Reid

I should attempt to treat human vice and folly geometrically ... the passions of hatred, anger, envy, and so on, considered in themselves, follow from the necessity and efficacy of nature ... I shall, therefore, treat the nature and strength of the emotion in exactly the same manner, as though I were concerned with lines, planes, and solids. — Baruch Spinoza

It appears that the solution of the problem of time and space is reserved to philosophers who, like Leibniz, are mathematicians, or to mathematicians who, like Einstein, are philosophers. — Hans Reichenbach

When I was young, most teachers of philosophy in British and American universities were Hegelians, so that, until I read Hegel, I supposed there must be some truth to his system; I was cured, however, by discovering that everything he said on the philosophy of mathematics was plain nonsense. — Bertrand Russell

Mere subtlety may qualify you as a sceptic but not as a philosopher. On the other hand, scepticism is in philosophy what the Opposition is in Parliament; it is just as beneficial, and indeed necessary. It rests everywhere on the fact that philosophy is not capable of producing the kind of evidence mathematics produces. — Arthur Schopenhauer

In trying to explain life we have reduced it to a series of chemical reactions, whether it be the burning of glucose in mitochondria to create energy, or the folding of proteins to make bile, or pollen, or blood. Zoom out to where we perceive things, the titanic mathematics of it all is silent. We have twisted our thoughts and feelings into all sorts of psychological origami about whether these things are a result of evolution, intelligent design, or creation ex nihilo, and for all we know, our little planet is the only place that holds all of this wonder in a void that is too staggeringly huge to conceive. — Sean J Halford

The science of government it is my duty to study, more than all other sciences; the arts of legislation and administration and negotiation ought to take the place of, indeed exclude, in a manner, all other arts. I must study politics and war, that our sons may have liberty to study mathematics and philosophy. Our sons ought to study mathematics and philosophy, geography, natural history and naval architecture, navigation, commerce and agriculture in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry and porcelain. — John Adams

Philosophers and psychiatrists should explain why it is that we mathematicians are in the habit of systematically erasing our footsteps. Scientists have always looked askance at this strange habit of mathematicians, which has changed little from Pythagoras to our day. — Gian-Carlo Rota

I earned my Ph.D. in philosophy, and one of my specializations was the logic and mathematics of game theory. I've also got a degree in drama, so I know about stories, characterizations, plot arcs, and the like. Lots of game designers can do one or the other: I've got the skills for both. — Brendan Myers

Ever since Plato most philosophers have considered it part of their business to produce 'proofs' of immortality and the existence of God. They have found fault with the proofs of their predecessors - Saint Thomas rejected Saint Anselm's proofs, and Kant rejected Descartes' - but they have supplied new ones of their own. In order to make their proofs seem valid, they have had to falsify logic, to make mathematics mystical, and to pretend that deepseated prejudices were heaven-sent intuitions. — Bertrand Russell

We do not learn by inference and deduction and the application of mathematics to philosophy, but by direct intercourse and sympathy. — Richard M. Nixon

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

Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. [ ... ] Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate. — Bertrand Russell

Man is a fantastic animal; he was born of fantasy, he is the son of "the mad woman of the house." And universal history is the gigantic and thousand-year effort to go on putting order into that huge, disorderly, anti-animal fantasy. What we call reason is no more than fantasy put into shape. Is there anything in the world more fantastic than that which is the most rational? Is there anything more fantastic than the mathematical point, and the infinite line, and, in general, all mathematics and all physics? Is there a more fantastic fancy than what we call "justice" and the other thing that we call "happiness"? — Jose Ortega Y Gasset

Art is a form of understanding like philosophy and science and mathematics are an understanding but the difference is that art has the capacity to hold all these different things. It is the form of understanding that is best suited for the contemporary time. — Robert Longo

There is one thing only which a Muslim can profitably learn from the west, the exact sciences in their pure and applied form. Only natural sciences and mathematics should be taught in Muslim schools, while tuition of European philosophy, literature and history should lose the position of primacy which today it holds on the curriculum. — Muhammad Asad

I went to Princeton from Amherst, where I split my interests between mathematics and philosophy. — Stephen Cole Kleene

I hold all knowledge that is concerned with things that actually exist - all that is commonly called Science - to be of very slight value compared to the knowledge which, like philosophy and mathematics, is concerned with ideal and eternal objects, and is freed from this miserable world which God has made. — Bertrand Russell

Men are constantly attracted and deluded by two opposite charms: the charm of competence which is engendered by mathematics and everything akin to mathematics, and the charm of humble awe, which is engendered by meditation on the human soul and its experiences. Philosophy is characterized by the gentle, if firm, refusal to succumb to either charm. — Leo Strauss

White folks was in the caves while we [blacks] was building empires ... We built pyramids before Donald Trump ever knew what architecture was ... we taught philosophy and astrology and mathematics before Socrates and them Greek homos ever got around to it. — Al Sharpton

I was perplexed by the failure of teachers at school to address what seemed the most urgent matter of all: the bewildering, stomach-churning insecurity of being alive. The standard subjects of history, geography, mathematics, and English seemed perversely designed to ignore the questions that really mattered. As soon as I had some inkling of what 'philosophy' meant, I was puzzled as to why we were not taught it. And my skepticism about religion only grew as I failed to see what the vicars and priests I encountered gained from their faith. They struck me either as insincere, pious, and aloof or just bumblingly good-natured. (p. 10) — Stephen Batchelor

I enjoyed mathematics from a very young age. At the beginning of college, I had this illusion, which was kind of silly in retrospect, that if I just understood math and physics and philosophy, I could figure out everything else from first principles. — Erez Lieberman Aiden

The membership relation for sets can often be replaced by the composition operation for functions. This leads to an alternative foundation for Mathematics upon categories
specifically, on the category of all functions. Now much of Mathematics is dynamic, in that it deals with morphisms of an object into another object of the same kind. Such morphisms (like functions) form categories, and so the approach via categories fits well with the objective of organizing and understanding Mathematics. That, in truth, should be the goal of a proper philosophy of Mathematics. — Saunders Mac Lane

Bits also play a part in logic, that strange blend of philosophy and mathematics for which a primary goal is to determine whether certain statements are true or false. True — Charles Petzold

Pure analysis puts at our disposal a multitude of procedures whose infallibility it guarantees; it opens to us a thousand different ways on which we can embark in all confidence; we are assured of meeting there no obstacles; but of all these ways, which will lead us most promptly to our goal? Who shall tell us which to choose? We need a faculty which makes us see the end from afar, and intuition is this faculty. It is necessary to the explorer for choosing his route; it is not less so to the one following his trail who wants to know why he chose it. — Henri Poincare

Sandor Boatly had never guessed that, properly played, baseball consisted of mathematics, geometry, art, philosophy, ballet, and carnival, all intertwined like the mystical ribbons of color in a rainbow. — W.P. Kinsella

After two years of undergraduate study, it was clear that I was bored by the regime of problem-solving required by the Cambridge mathematical tripos. A very sensitive mathematics don recommended that I talk to the historian of astronomy, Michael Hoskin, and the conversation led me to enroll in the History and Philosophy of Science for my final undergraduate year. — Philip Kitcher

I started studying law, but this I could stand just for one semester. I couldn't stand more. Then I studied languages and literature for two years. After two years I passed an examination with the result I have a teaching certificate for Latin and Hungarian for the lower classes of the gymnasium, for kids from 10 to 14. I never made use of this teaching certificate. And then I came to philosophy, physics, and mathematics. In fact, I came to mathematics indirectly. I was really more interested in physics and philosophy and thought about those. It is a little shortened but not quite wrong to say: I thought I am not good enough for physics and I am too good for philosophy. Mathematics is in between. — George Polya

The Pythagoreans, you have to remember, were extremely weird. Their philosophy was a chunky stew of things we'd now call mathematics, things we'd now call religion, and things we'd now call mental illness. — Jordan Ellenberg

Like mathematics and music and cosmology and philosophy, poetry, too, can "infinitize" us, granting us what immortality there is to be had in this mortal life. And all those who vibrate in harmony to language that itself vibrates to the harmonies of the infinite are entitled to inclusion among the "small group of people. — Rebecca Goldstein

Can you name a single one of the great fundamental and original intellectual achievements which have raise man in the scale of civilization that may be credited to the Anglo-Saxon? The art of letters, of poetry, of music, of sculpture, of painting, of the drama, of architecture; the science of mathematics, of astronomy, of philosophy, of logic, of physics, of chemistry, the use of the metals and principles of mechanics, were all invented or discovered by darker and what we now call inferior races and nations. — James Weldon Johnson

Science in its everyday practice is much closer to art than to philosophy. When I look at Godel's proof of his undecidability theorem, I do not see a philosophical argument. The proof is a soaring piece of architecture, as unique and as lovely as Chartres Cathedral. Godel took Hilbert's formalized axioms of mathematics as his building blocks and built out of them a lofty structure of ideas into which he could finally insert his undecidable arithmetical statement as the keystone of the arch. The proof is a great work of art. It is a construction, not a reduction. It destroyed Hilbert's dream of reducing all mathematics to a few equations, and replaced it with a greater dream of mathematics as an endlessly growing realm of ideas. Godel proved that in mathematics the whole is always greater than the sum of the parts. Every formalization of mathematics raises questions that reach beyond the limits of the formalism into unexplored territory. — Freeman Dyson

But ignorance of divine revelation affects all of thought and life, from one's view toward history and philosophy, to one's interpretation of music and literature, to one's understanding of mathematics and physics. — Vincent Cheung

There is no "religious language" or "scientific language". There is rather the international notation of mathematics and logic; and English, French, Spanish and the like. In short, "religious discourse" and "scientific discourse" are part of the same overall conceptual structure. Moreover, in that conceptual structure there is a large amount of discourse, which is neither religious nor scientific, that is constantly being utilized by both the religious man and the scientist when they make religious and scientific claims. In short, they share a number of key categories. — Kai Nielsen

Only in mathematics and physics was I, through self-study, far beyond the school curriculum, and also with regard to philosophy as it was taught in the school curriculum. — Albert Einstein

Primary causes are unknown to us; but are subject to simple and constant laws, which may be discovered by observation, the study of them being the object of natural philosophy.
Heat, like gravity, penetrates every substance of the universe, its rays occupy all parts of space. The object of our work is to set forth the mathematical laws which this element obeys. The theory of heat will hereafter form one of the most important branches of general physics. — Joseph Fourier

I had a liberal arts education at Amherst College where I had two majors, mathematics and philosophy. — Stephen Cole Kleene

The evident character of this defective cognition of which mathematics is proud, and on which it plumes itself before philosophy, rests solely on the poverty of its purpose and the defectiveness of its stuff, and is therefore of a kind that philosophy must spurn — Georg Wilhelm Friedrich Hegel

No code of ethics and no effort are justifiable a priori in the face of the cruel mathematics that command our condition. — Albert Camus

Perhaps, you know, new laws, new domains of potential openness are occurring as the universe ages, and complexity previously disallowed is now possible, and we are that complexity. We are nature moving out of its genetic phase - a phase under the control of chemical genes, which are physical structures, in to an epigenetic phase, a phase of culture ruled by codes, transformable culturally confined codes - mathematics, religion, philosophy, art, dance, humor. — Terence McKenna

The three main medieval points of view regarding universals are designated by historians as realism, conceptualism, and nominalism. Essentially these same three doctrines reappear in twentieth-century surveys of the philosophy of mathematics under the new names logicism, intuitionism, and formalism. — Willard Van Orman Quine

The true man of science will know nature better by his finer organization; he will smell, taste, see, hear, feel, better than other men. His will be a deeper and finer experience. We do not learn by inference and deduction and the application of mathematics to philosophy, but by direct intercourse and sympathy. It is with science as with ethics,
we cannot know truth by contrivance and method; the Baconian is as false as any other, and with all the helps of machinery and the arts, the most scientific will still be the healthiest and friendliest man, and possess a more perfect Indian wisdom. — Henry David Thoreau

From Pythagoras (whether by way of Socrates or not) Plato derived the Orphic elements in his philosophy: the religious trend, the belief in immortality, the other-worldliness, the priestly tone, and all that is involved in the simile of the cave; also his respect for mathematics, and his intimate intermingling of intellect and mysticism. — Bertrand Russell

The dairy man had a Ph.D. in mathematics, and he must have had some training in philosophy. He liked what he was doing and he didn't want to be somewhere else - one of the few contented people I met in my whole journey. — John Steinbeck

I must study politics and war that my sons may have liberty to study mathematics and philosophy. My sons ought to study mathematics and philosophy, geography, natural history, naval architecture, navigation, commerce, and agriculture in order to give their children a right to study paintings, poetry, music, architecture, statuary, tapestry, and porcelain. How — David McCullough

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. — Jules Michelet

The world of being is unchangeable, rigid, exact, delightful to the mathematician, the logician, the builder of metaphysical systems, and all who love perfection more than life. The world of existence is fleeting, vague, without sharp boundaries, without any clear plan or arrangement, but it contains all thoughts and feelings, all the data of sense, and all physical objects, everything that can do either good or harm, everything that makes any difference to the value of life and the world. According to our temperaments, we shall prefer the contemplation of the one or of the other. — Bertrand Russell

The focus of history and philosophy of science scholar Arthur Miller's (2010) "137: Jung and Pauli and the Pursuit of Scientific Obsession" is Jung and Pauli's
mutual effort to discover the cosmic number or fine structure constant, which is a fundamental physical constant dealing with electromagnetism, or, from a different perspective, could be considered the philosopher's stone of the mathematical universe.
This was indeed one of Pauli and Jung's collaborative passions, but it was not the only concentration of their relationship. Quantum physics could be seen as the natural progression from ancient alchemy, through chemistry, culminating in the abstract world of subatomic particles, wave functions, and mathematics. [Ancient Egypt and Modern Psychotherapy] — Todd Hayen

The person who wishes to attain human perfection should study logic first, next mathematics, then physics, and, lastly, metaphysics. — Maimonides

We have heard much about the poetry of mathematics, but very little of it has as yet been sung. The ancients had a juster notion of their poetic value than we. The most distinct and beautiful statements of any truth must take at last the mathematical form. We might so simplify the rules of moral philosophy, as well as of arithmetic, that one formula would express them both. — Henry David Thoreau

Sciences are differentiated according to the various means through which knowledge is obtained. For the astronomer and the physicist both may prove the same conclusion: that the earth, for instance, is round: the astronomer by means of mathematics (i.e. abstracting from matter), but the physicist by means of matter itself. Hence there is no reason why those things which may be learned from philosophical science, so far as they can be known by natural reason, may not also be taught us by another science so far as they fall within revelation. Hence theology included in sacred doctrine differs in kind from that theology which is part of philosophy. SECOND ARTICLE [I, Q. — Thomas Aquinas

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

Logic issues in tautologies, mathematics in identities, philosophy in definitions; all trivial, but all part of the vital work of clarifying and organising our thought. — Frank Plumpton Ramsey

All the sciences have a relation, greater or less, to human nature; and ... however wide any of them may seem to run from it, they still return back by one passage or another. Even Mathematics, Natural Philosophy, and Natural Religion, are in some measure dependent on the science of MAN; since they lie under the cognizance of men, and are judged of by their powers and faculties. — David Hume

University President: Why is it that you physicists always require so much expensive equipment? Now the Department of Mathematics requires nothing but money for paper, pencils, and erasers ... and the Department of Philosophy is better still. It doesn't even ask for erasers. — Isaac Asimov

All those formal systems, in mathematics and physics and the philosophy of science, which claim to give foundations for certain truth are surely mistaken. I am tempted to say that we do not look for truth, but for knowledge. But I dislike this form of words, for two reasons. First of all, we do look for truth, however we define it, it is what we find that is knowledge. And second, what we fail to find is not truth, but certainty; the nature of truth is exactly the knowledge that we do find. — Jacob Bronowski

But Miss Ferguson preferred science over penmanship. Philosophy over etiquette. And, dear heavens preserve them all, mathematics over everything. Not simply numbering that could see a wife through her household accounts. Algebra. Geometry. Indecipherable equations made up of unrecognizable symbols that meant nothing to anyone but the chit herself. It was enough to give Miss Chase hives.

The girl wasn't even saved by having any proper feminine skills. She could not tat or sing or draw. Her needlework was execrable, and her Italian worse. In fact, her only skills were completely unacceptable, as no one wanted a wife who could speak German, discuss physics, or bring down more pheasant than her husband. — Eileen Dreyer

Finally, the dishonesty in the movement of the publication of a Greek philosophy, becomes very glaring, when we refer to the fact, purposely that by calling the theorem of the Square on the Hypotenuse, the Pythagorean theorem, it has concealed the truth for centuries from the world, who ought to know that the Egyptians taught Pythagoras and the Greeks, what mathematics they knew. — George G. M. James

We the undersigned, intend to establish an instruction and training institution which differs from the common elementary schools principally in that it will embrace, outside of (in addition to) the general and elementary curriculum, all branches of the classical high school, which are necessary for a true Christian and scientific education, such as: Religion, the Latin, Greek, Hebrew, German, French and English languages; History, Geography, Mathematics, Physics, natural history, Introduction to Philosophy, Music, and Drawing. — C.F.W. Walther

I received my high school baccalaureate diploma in Latin and Science in 1928, then my two baccalaureate diplomas in Mathematics and Philosophy in 1929. — Maurice Allais

I am too good for philosophy and not good enough for physics. Mathematics is in between. — George Polya

Histories make men wise; poets, witty; the mathematics, subtle; natural philosophy, deep; moral, grave; logic and rhetoric, able to contend. — Francis Bacon

The analytical geometry of Descartes and the calculus of Newton and Leibniz have expanded into the marvelous mathematical method-more daring than anything that the history of philosophy records-of Lobachevsky and Riemann, Gauss and Sylvester. Indeed, mathematics, the indispensable tool of the sciences, defying the senses to follow its splendid flights, is demonstrating today, as it never has been demonstrated before, the supremacy of the pure reason. — Nicholas Murray Butler

Academics are only useful when they try to be useless (say, as in mathematics and philosophy) and dangerous when they try to be useful. — Nassim Nicholas Taleb

Surely, the gods' judgment is certain. But as for us, we must be satisfied to 'come close' to those things, for we are men, who speak according to what is likely, and whose lectures resemble fables. — Proclus

The question as to which of these two theories applies to the actual world is, like all questions concerning the actual world, in itself irrelevant to pure mathematics.* But the argument against absolute position usually takes the form of maintaining that a space composed of points is logically inadmissible, and hence issues are raised which a philosophy of mathematics must discuss. In what follows, I am concerned only with the question: Is a space composed of points self-contradictory? It is true that, if this question be answered in the negative, the sole ground for denying that such a space exists in the actual world is removed; but this is a further point, which, being irrelevant to our subject, will be left entirely to the sagacity of the reader. — Bertrand Russell

I do not think the division of the subject into two parts - into applied mathematics and experimental physics a good one, for natural philosophy without experiment is merely mathematical exercise, while experiment without mathematics will neither sufficiently discipline the mind or sufficiently extend our knowledge in a subject like physics. — Balfour Stewart

In mathematics or physics, infinity is greater than one or two or any number countable. In how many ways can the world be destroyed based on ordered knowledge? You may be able to count this. But the truth is, you "really" don't know. These possibilities in your mind hold a set of unpredictable orders. One effect may be causative of another of another. It could be a culmination of effects you know as events where events are sets and subsets of potential possibilities.
In how many ways can the world be destroyed based on unordered possibilities? — Dew Platt

See skulking Truth to her old cavern fled, Mountains of Casuistry heap'd o'er her head! Philosophy, that lean'd on Heav'n before, Shrinks to her second cause, and is no more. Physic of Metaphysic begs defence, And Metaphysic calls for aid on Sense! See Mystery to Mathematics fly! — Alexander Pope

Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these one is wandering in a dark labyrinth. - Galileo Galilei, The Assayer, 1623 — Max Tegmark

Even in the realm of pure mathematics, the mathematician may use any set of symbols he desires within any given region of space-time; he may even go so far as to maintain that any one set of symbols fits the scheme as well as any other, but to erect this method into a philosophy and confuse independence of any one special meaning with independence of all meaning is unjustified and unwarranted. — Fulton J. Sheen