# How can I think deep math

## Quotes on math

So far it has not been proven that anything is difficult in mathematics.

Norbert a'Campo

High technology is essentially mathematical technology.

Study Commission of the American Academy of Sciences

Please forget everything you learned in school; because you haven't learned

Edmund LandauYou don't learn math, you just get used to it.

Paul ErdösLearning math means reinventing it over and over again.

Donal O'SheaIt is also true of mathematical calculation that it is better to understand it than to master it.

Hanspeter RingsA mathematical truth in itself is neither simple nor complicated, it is.

Émile Lemoine

As strange as it may sound, the strength of mathematics rests on the avoidance of unnecessary assumptions and on its great savings in thinking.

Ernst Mach, physicist

Because even thinking is sometimes harmful to health.

Aristotle

A mathematician who is not in some way a poet will never be a perfect mathematician.

Karl Weierstrasse

Nos mathematici sumus isti veri poetae sed quod fingimus nos et probare decet. (We mathematicians are the real poets, we just have to prove what our imagination can do.)

Leopold Kronecker

Mathematics is the music of reason.

James Joseph Sylvester

Mathematics is the music of the mind, music is the mathematics of the soul.

Daniil Charms (1905-1942)

The preoccupation with mathematics educates to objective thinking, it defends the inadmissible generalization, it brings about a precision of the language.

Herbert Meschkowski

Mathematics as a subject is so serious that no opportunity should be missed to make it more entertaining.

Blaise Pascal

A good math joke is always better than a whole dozen mediocre scholarly treatises.

John Edensor Littlewood

Funny examples sometimes carry more weight than serious ones.

Michael Stifel

Squaring the circle: You don't know what it is, but shrug your shoulders when the conversation comes up.

Gustav Flaubert (1821-1880)

Mathematics is more of an activity than a teaching.

L. E. J. Brouwer

The best of all languages in the world is an artificial language, a fairly compact language, the language of mathematics.

N. I. Lobachevsky

The book of nature is written with mathematical symbols.

More precisely: nature speaks the language of mathematics: the letters of this language are triangles, circles and other mathematical figures.

Galileo Galilei

Whoever understands geometry can understand everything in this world.

Galileo Galilei

Measure everything that can be measured and make everything measurable that cannot be measured.

Galileo Galilei

It is impossible to adequately convey the beauties of the laws of nature if one does not understand math. I regret that, but I guess it is.

Richard Feynman, 1965 Nobel Laureate in Physics

There are things that seem incredible to most people who haven't studied math.

Archimedes

Mathematics is a kind of toy that nature throws at us for comfort and entertainment in the dark.

Jean-Baptist le Rond d'Alembert

The imagination works no less in a creative mathematician than in an inventive poet.

Jean-Baptist le Rond d'Alembert

Mathematics is the most pleasant science; she and astronomy represent dance companies, concerts, and other such amusements for me, which I only know by name.

Karl Friedrich Bessel (18 years old)

Not that the more widespread insight into the method of mathematics would necessarily mean that much more wise things would be said than it is today, but that much less unwise things would certainly be said.

Karl Menger

Religion and mathematics are just different expressions of the same divine accuracy.

Cardinal Michael Faulhaber (1869-1952)

Medicine makes people sick, mathematics sad, and theology sinful.

Martin Luther

So mathematics can be defined as the science in which we never know what we are talking about and never know whether what we are saying is true.

Bertrand Russell

Since the simplest of claims began to be proven, many of them have been proven false.

Bertrand Russell

Some people have a zero radius of view and call it their point of view.

David Hilbert

Like no other question, the infinite has always moved man's mind so deeply; The infinite has had such a stimulating and fruitful effect on the understanding as hardly any other idea; But the infinite is in need of enlightenment like no other concept.

David Hilbert

I know that geometry was the first where I got to know happiness.

Rudolf Steiner (1861-1925)

In the large garden of geometry, everyone can pick a bouquet according to their taste.

David Hilbert

Mathematics is the instrument that mediates between theory and practice, between thinking and observing: it builds the connecting bridge and makes it ever more sustainable. Hence it is that our whole contemporary culture, in so far as it is based on the spiritual penetration and utilization of nature, finds its basis in mathematics.

David Hilbert

There is the problem, find the solution; You can find them through pure thinking; because there is no ignorance in mathematics.

David Hilbert

Fear of mathematics is much closer to fear than to awe.

Felix Auerbach

The majority have feelings towards mathematics as, according to Aristotle, they should be awakened by tragedy, namely pity and fear. Pity for those who have to struggle with mathematics, and fear that one might find oneself in this dangerous position one day.

Paul Epstein (1883-1966)

Archimedes will be remembered when Aeschylus is forgotten - because languages die, but mathematical ideas do not.

Godfrey Harold Hardy

The decisive criterion is beauty; There is no permanent place in this world for ugly mathematics.

Godfrey Harold Hardy

Studying mathematics, I say, is the best remedy for cupidity.

Thomas Mann (Councilor Behrens in: The Magic Mountain)

Mathematics is not detrimental to the love instinct.

Paul Möbius (1853-1907), psychiatrist (not: August Möbius)

The good Christian should beware of mathematicians and all those who make empty predictions, especially when these predictions are correct. There is a danger that mathematicians in league with the devil will cloud the spirit and become entangled in the bonds of hell.

Augustine

(However, mathematician means astrologer here. Today's mathematicians were called geometers at the time: learning the art of geometry and doing it publicly is of value, but the damnable mathematical art is forbidden. Under Roman law)

Mathematics, like godliness, is useful for all things, but like this, not for everyone.

Jakob Kraus (1753-1807)

At least I don't know of a fully satisfactory explanation why every odd number (from 3), multiplied by itself, always results in a multiple of 8 with 1 as the remainder.

Erich Bischoff, researcher of Kabbalah, 1920

It is unbelievable how ignorant the young students at universities get when I calculate or geometrize only 10 minutes, 1/4 of them gently fall asleep.

Georg Christoph Lichtenberg (1742-1799)

I believe that, in the strictest sense, there is only one science for man, and that is pure mathematics. For this we need nothing more than our mind.

Georg Christoph Lichtenberg (1742-1799)

The so-called mathematicians by profession, based on the immaturity of the rest of the world, have acquired a profound credit which is very similar to that of holiness which theologians have for themselves.

Georg Christoph Lichtenberg (1742-1799)

Mathematics is a wonderful science, but mathematicians are often unsuitable for the hangman. It's almost like mathematics like theology. Just as the latter assiduous, especially when they are in office, claim a special credit of holiness and a closer relationship with God, although many of them are really good-for-nothing, so very often the so-called mathematician, who is considered a deep thinker, demands to do so Even if there are the greatest junkheads that can be found right below them, they will become unsuitable for any business that requires reflection, unless it can be done directly through that easy combination of signs, which are more the work of routine than of thought .

Georg Christoph Lichtenberg (1742-1799)

There is no greater obstacle to progress in the sciences than the need to feel the success of it too soon.

Georg Christoph Lichtenberg (1742-1799)

Morality has more value to society than any Newtonian calculation.

Frederick the Great (1712 - 1786)

This is a means not to miss Paradise: on the one hand a mathematician, on the other a Jesuit; with this company you have to make your way, or you never make it.

Frederick the Great (1712 - 1786)

If we cannot approach the divine in any other way than through symbols, we will use mathematical symbols most appropriately, for these have indestructible certainty. Knowledge of the divine is unattainable for a mathematically completely uneducated person.

Nicholas of Cues

God is a child and when he started playing he was doing math. It is the most divine gimmick among humans.

V. Erath

Anything that is just likely is likely wrong.

Rene Descartes

Of all those who have so far searched for the truth, mathematicians alone have been able to find a number of proofs, from which it follows that their subject must have been the easiest of all.

Rene Descartes

Mathematics is the science of what goes without saying.

Heinrich Heine (1797-1856)

The mathematicians, who are only mathematicians, think correctly, but only on condition that all things are explained to them by definitions and principles; otherwise they are limited and unbearable, because they only think correctly when it comes to very clear principles.

Blaise Pascal

One can have three main goals in studying the truth: once, to discover it, if one seeks it; then: to prove it when one has it; and lastly: to distinguish them from falsehood when you examine them.

Blaise Pascal

There is no more harm in the truth than by trying to base it on false conclusions.

P. L. M. de Maupertuis

Mathematical definitions can never be wrong.

Imamnuel Kant (1724-1804)

In any pure doctrine of nature there is only so much actual science contained in it as mathematics can be applied in it.

Imamnuel Kant (1724-1804)

(Almost literally found in Leonardo da Vinci)

Those who blame the sublime wisdom of mathematics feed on confusion.

Leonardo da Vinci

We want to drive the subtlety and rigor of mathematics into all sciences as far as this is possible, not in the belief that we will recognize things in this way, but in order to determine our human relation to things. Mathematics is only the means of general and ultimate knowledge of human nature.

Friedrich Wilhelm Nietzsche (1844-1900)

The invention of the laws of numbers was made on the basis of the originally already prevailing error that there are several identical things (but in fact there is nothing the same), at least that there are things (but there is no "thing").

Friedrich Wilhelm Nietzsche (1844-1900)

One of the greatest facts in the history of mind was mathematics.

Jacob Christoph Burckhardt (1818-1897)

Insofar as the propositions of mathematics relate to reality, they are not certain, and insofar as they are certain they do not relate to reality.

Albert Einstein

How is it possible that mathematics, which is ultimately a product of human thought independent of experience, corresponds so wonderfully to the real situation?

Albert Einstein

Since the mathematicians attacked the theory of relativity, I have not understood it myself anymore.

Albert Einstein

Don't worry about your math difficulties. I can ensure you, mine are even bigger.

Albert Einstein

God doesn't care about our math difficulties. He integrates empirically.

Albert Einstein

Equations are more important to me because politics is for the present, but an equation for the ages.

Albert Einstein

That whereby our calculations fail is what we call coincidence.

Albert Einstein

The only area that knows no coincidence is mathematics. Maybe that's why it's so utterly boring and unpopular.

Stefan Wittlin, Swiss "Medicus-Canis"

My former math teacher died. How I would have loved to tell him that about 40 years ago I was correct in my assertion that I would definitely never need algebra.

Stefan Wittlin, Swiss "Medicus-Canis"

The scientist finds his reward in what Poincaré calls the joy of understanding, not in the possible applications of his invention.

Albert Einstein

The only way to improve our conclusions is to make them as vivid as those of mathematicians, in such a way that one can find one's error with one's eyes and, when there are disputes among people, one need only say: " Let's do the math! " without any further formality to see who is right.

G. W. Leibniz

All educators agree that one must, above all, do a good amount of mathematics, because knowledge of it is of the greatest direct benefit in life.

Felix Klein

We mathematicians and physicists are proud to have the knowledge that we call our own a field of knowledge which is progressively offering mankind ever new external success and inner insight, and we must and want this joy in our possession if it is ever lost should have gone regain!

Felix Klein

It cannot be denied that much of elementary mathematics is of considerable practical use. But those pieces of math are pretty boring overall. These are the very parts of math that have the least aesthetic value. The "real" mathematics of the "real" mathematicians, the mathematics of Fermat, Gauss, Abel and Riemann, is almost completely "useless".

Godefrey Harold Hardy

He's a mathematician, so he's persistent.

Johann Wolfgang von Goethe (1749-1832)

There is no cheerful relationship to be won with mathematicians.

Johann Wolfgang von Goethe (1749-1832)

Mathematicians are a kind of French: if you talk to them, they translate it into their language, and then it is immediately something completely different.

Johann Wolfgang von Goethe (1749-1832)

I heard myself accusing myself as if I were an adversary, an enemy of mathematics in general, which nobody can appreciate more than me, since it does exactly what I have been completely refused to do.

Johann Wolfgang von Goethe (1749-1832)

Mathematics has a very wrong reputation for delivering unmistakable conclusions.

Johann Wolfgang von Goethe (1749-1832)

Well, the investigation would finally be wrapped in the secrets of mathematics, so that nobody would dare to approach this sanctuary so easily.

Johann Wolfgang von Goethe (1749 - 1832) (on Newton's "Optics")

The mathematical force is the ordering force.- The concept of mathematics is the concept of science in general. All sciences should therefore become mathematics - The highest life is mathematics - The life of the gods is mathematics - Pure mathematics is religion - Anyone who does not take a mathematical book with devotion and read it like God's word does not understand it. All divine messengers must be mathematicians.

Novalis, who by the way studied at the Bergakademie Freiberg!

(Though thinks *mathematics* number mysticism here!)

God exists because the math is consistent, and the devil exists because we can't prove it.

Andre Weil

The good God created the whole numbers, everything else is human work.

Leopold Kronecker

(Helmut Hasse therefore wrote in his book *Lectures on number theory* in the list of authors under L the dear God listed!)

There is no silver bullet to mathematics.

Euclid

Structures are the weapons of mathematicians.

N. Bourbaki

Since the time of the Greeks, "mathematics" means to say "proof".

N. Bourbaki

His brain is growing all the time, at least for the parts that deal with math. They swell up bigger and bigger ...His voice becomes a mere snarling, enough to reproduce formulas. He loses the ability to laugh unless it is the sudden discovery of a paradoxical problem. He feels most moved when he solves a new arithmetic problem.

H. G. Wells (1866-1946)

Up to this hour I am not familiar with the multiplication tables.

Franz Grillparzer (1791 - 1872)

Arithmetic is not an opinion.

Bernardino Grimaldi (1841-1897)

I have no faith in the arithmetic of politics.

Adam Smith (1723-1790)

Lotteries and the lottery are the late punishments for people who failed to pay attention to math while in school.

Willy Meurer

Anyone who cannot think of a point is simply too lazy to do so.

Maths teacher Brenneke in "Eduards Traum" by Wilhelm Busch (1832-1908)

The math point is an angle that both legs have been torn out.

Johann Georg August Galetti (1750-1828)

That the lowest of all activities is arithmetic is shown by the fact that it is the only one that can also be carried out by a machine. But now all analysis finitorum et infinitorum basically goes back to calculation. Then measure the "mathematical profundity".

Arthur Schopenhauer (1788-1860)

Here, too, the basic principle is an unjustified application of a logical method to cases that, strictly speaking, should not be subsumed under it, or the consideration of such structures as numbers that are not right numbers at all. Negative numbers are a self-contradiction, as all mathematicians admit.

Hans Vaihinger (1852-1933)

1 * 1 = 1, undoubtedly. But is not 1 because the square of a given number has to be larger than the number itself. Logically, the root of 1 cannot be 1 because the root of a number has to be smaller than the number itself. But is mathematical or formal. Mathematics in this case contradicts logic or pure reason, and therefore mathematics in this cardinal case is irrational. All values are then based on this senselessness, the 1, and mathematical science, the "only exact, infallible", is based on these false values. But this is math! Nice game for people who have nothing to do.

August Strindberg (1849-1912)

At the beginning of this century, a self-destructive democratic principle was introduced into mathematics (especially by Hilbert), according to which all axiom systems have the same right to analysis and the value of a mathematical achievement is determined not by its importance and its usefulness for other disciplines, but just by its difficulty, as in mountaineering. This principle quickly led mathematicians to break with physics and isolate themselves from all other sciences. In the eyes of all normal people they turned into an obscure priestly caste ... Strange questions like Fermat's problem or sums of prime numbers were raised to supposedly central problems of mathematics.

Vladimir Igorewitsch Arnold (born 1937)

I have the bad habit of taking a lively interest in mathematical subjects only where I can suspect meaningful combinations of ideas and results that are recommendable through elegance or generality.

Carl Friedrich Gauss

One must not confuse what seems improbable and unnatural to us with what is absolutely impossible.

Carl Friedrich Gauss

The lack of mathematical education is not evident in anything as strikingly as through excessive sharpness in numerical calculations.

Carl Friedrich Gauss

I already have the result, now I just need the path that leads to it.

Carl Friedrich Gauss

Mathematics is a wonderful teacher of the art of organizing thoughts, eliminating nonsense and creating clarity.

J. H. Fabre

One must study mathematics because it organizes thoughts.

M. W. Lomonosov

You wanted algebra, there you have the salad.

Jules Verne, Journey around the Moon, Chapter 4 (1828-1905)

I have to prove this cheese

otherwise the work is dubious.

F. Will

A math problem can sometimes be as fun as a crossword puzzle, and tense mental work can be as desirable an exercise as a quick game of tennis.

George Polya

Don't forget: if you want to learn to swim, go boldly into the water; if you want to learn to solve problems, then solve them.

George Polya

One must guess a mathematical proposition before proving it, one must guess the idea of a proof before going into the details.

George Polya

This principle is so perfectly general that it cannot be applied.

George Polya

The first rule to follow in math is to be exact. The second rule is to be clear and precise and, if possible, simple.

Lazare Nicolas Marguerite Carnot

There are no authorities in mathematics. The only argument for the truth is evidence.

K. Urbanik

Logic is the hygiene that mathematicians use to keep their thoughts healthy and vigorous.

Hermann Weyl

Thinking is one of the greatest pleasures of the human race.

Bertolt Brecht

No one learns to think by reading the finished thoughts of others, but by thinking for himself.

M. Eminescij

The kind of knowledge that is only supported by observation and not yet proven must be carefully distinguished from the truth; it is, as we usually say, obtained by induction. Yet we have seen cases where mere induction has led to error. We should therefore take great care not to accept as true such numerical properties which we have discovered by observation and which are supported by induction alone.

Leonhard Euler

For this purpose, content is enclosed in algebraic formulas so that by using the formula one does not have to repeat the same thing a hundred times.

A. I. Hearts

The rules of arithmetic established thousands of years ago hold their power in the era of modern mathematics and are used literally at every step.

B. W. Gnedenko

You may add up as many zeros as you want: there is no one.

Galileo Galilei

Of all human discoveries, the most important thing should be the discovery of the faults.

Stanislaw Jerzy Lec

I do not agree with the math. I think the sum of zeros is a dangerous number.

Stanislaw Jerzy Lec

Mathematics already teaches us that zeros must not be overlooked.

Gabriel Laub (1928-1998)

After the big bank crash, the zeros count more than any other number.

Willi Meurer

It is worth paying attention in school, because mathematics teaches that & rsquo; one cannot overlook zeros. No other insight pays off so well in career planning.

Karl-Heinz Karius

Of all that is accessible to the human mind, nothing is more pleasant and worthy than the knowledge of the truth.

G. Cardano

Truth has to overcome a thousand obstacles in order to get to paper undamaged and from paper to head again.

Georg Christoph Lichtenberg (1742-1799)

People who want to go the right way must also know about wrong ways.

Aristotle

Of all the true sciences, as Aristotle and Averroes maintain, the mathematical sciences are the truest and, in indisputability, come first over the natural sciences.

Luca Pacioli

Only in mathematics is there irrefutable evidence based on necessary premises. That is why man can only arrive at the truth in mathematics, based on the laws peculiar to this science.

Roger Bacon

Only in mathematics is there scientificity and evidential value in the most precise and proper sense.

Robert Grosseteste

Everything has to be proven, and in proving one must not use anything except axioms and previously proven propositions.

Blaise Pascal

In mathematical questions, even the smallest mistake must not be ignored.

Isaac Newton

People who do not know anything about algebra also cannot imagine the wonderful things that one can get with the help of said science.

Gottfried Wilhelm Leibniz

First and foremost, mathematics protects us from hallucinations and teaches us that it is one thing how things are constructed that we grasp with the senses, and another how they appear to us. This science gives us the most reliable rules, whoever is guided by them need not be afraid of hallucinations.

Leonhard Euler

A thousand ways lead to mistakes, only one leads to the truth.

Jean-Jacques Rousseau

What is true is neither good nor bad, it is simply true.

R. Tagore

He who refuses to engage in arithmetic is doomed to tell nonsense.

John McCarthy

'Obviously' is the most dangerous word in math.

Eric Temple Bell

Mathematical knowledge builds on the work of those who came before us. It's hard earned, and often we don't appreciate it the way we should. Anyone who has a normal school education today can solve arithmetic and algebraic problems at which most Babylonian scholars would have failed. And anyone with a few semesters of analysis and linear algebra can master problems that Pythagoras, Archimedes, and not even Newton could have cracked. With a degree in mathematics, you can now perform topological calculations that Riemann or Poincaré would not even have started. We are no smarter than them. We are their beneficiaries.

Donal O'Shea, in: Poincaré's Conjecture

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