Since mathematics is related to logic, he cites an example from mathematics: If we have a formula like ( a + b )( a-b )= a²-b² it does not tell us how to think mathematically.

If an isomorphism can be found from a relatively unknown part of mathematics into some well studied division of mathematics, where many theorems are already proved, and many methods are already available to find answers, then the function can be used to map whole problems out of unfamiliar territory over to " solid ground " where the problem is easier to understand and work with.

In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa.

If one agrees that set theory is an appealing foundation of mathematics, then all mathematical objects must be defined as sets of some sort.

If one of the two events represents the sending of a signal from one location and the second event represents the reception of the same signal at another location, then as long as the signal is moving at the speed of light or slower, the mathematics of simultaneity ensures that all reference frames agree that the transmission-event happened before the reception-event.

If mathematics is a part of logic, then questions about mathematical objects reduce to questions about logical objects.

If mathematics is a language, it is a different type of language than natural languages.

" ( e. g., " If Isaac Newton and Gottfried Leibniz had cooperated with each other, what would mathematics look like today?

Roger Bacon wrote " If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.

One famous apocryphal quote attributed to him upon hearing an explanation of the extremely complicated mathematics required to demonstrate Ptolemy's theory of astronomy was " If the Lord Almighty had consulted me before embarking on creation thus, I should have recommended something simpler.

If modern mathematical practices are what distinguish modern professional mathematicians from older ideas of folk mathematics.

If mathematics has been informally used throughout history, in numerous cultures and continents, then it could be argued that " mathematical practice " is the practice, or use, of mathematics in everyday life.

In mathematics, and in particular linear algebra, a skew-symmetric ( or antisymmetric or antimetric ) matrix is a square matrix A whose transpose is also its negative ; that is, it satisfies the equation If the entry in the and is a < sub > ij </ sub >, i. e. then the skew symmetric condition is For example, the following matrix is skew-symmetric:

If Göttingen had been the “ Mecca ” of mathematics in the 1920s and early ‘ 30s, Princeton, following the decimation of German mathematics under the Nazis, had become the center of the mathematical world in the 1940s.

In the novel, Faria, an Italian, is a prisoner of the Château d ' If who instructs Edmond Dantès, the protagonist and a fellow prisoner, in a number of fields including mathematics, the sciences, and foreign languages, and eventually helps him to escape from the island prison.

If God has mathematics of his own that needs to be done, let him do it himself.

In mathematics, the Sato – Tate conjecture is a statistical statement about the family of elliptic curves E < sub > p </ sub > over the finite field with p elements, with p a prime number, obtained from an elliptic curve E over the rational number field, by the process of reduction modulo a prime for almost all p. If N < sub > p </ sub > denotes the number of points on E < sub > p </ sub > and defined over the field with p elements, the conjecture gives an answer to the distribution of the second-order term for N < sub > p </ sub >.

He is also famous for having said, " If I feel unhappy, I do mathematics to become happy.

If I am happy, I do mathematics to keep happy.

If the consensus declared 2 + 2 = 5 it would render the practice of mathematics where 2 + 2 = 4 impossible.

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

If you tell him I made a pass at you he might think you misunderstood something I said or did, so instead of just telling him I made a pass, say I tried to date you and that you agreed so you could prove to him what a louse I really am.

If you're lyin' out in the hammock at night, and it gets kinda cool -- you know -- you just take these sides with the fringe on -- see -- and wrap 'em right over you.

If he snatches a toy, she says, `` Caroline wants her own truck just as you do ''.

If we go into this Common Market, we might just as well stop talking about Constitutional guarantees, Connally Amendments or, for that matter, conservatism in general.

If he were to go with White, he would be out there two days, not just listening in the dark at some point between here and Papa-san, but moving ever deeper into enemy land -- behind Papa-san -- itself.

If those aren't enough for you, why not grow some just for winter blooming??

If you wish to budget closely on transportation, saving your extra dollars to indulge in luxuries, one agency lists the small Fiat 500 at only $1.26 a day plus $.03 a kilometer and the Fiat 2100 Station Wagon, seating six, at just $1.10 a day and $.105 a kilometer.

If you drive greater distances than that, you'll just be skimming the surface and will never discover the enchantment, fascination and beauty which lured you in the first place to explore the hinterlands.

If this is the case, one would expect that not only the various procedures just mentioned which alter the hypothalamic balance would influence emotional state and behavior but that emotion itself would act likewise.

If asked how she gets her impressions, she probably can only say that she `` just gets them '' -- some more vividly than others.

If you just leave me to hell alone, Lieutenant.

If the acting captain wanted his acting lieutenant to sit on his ass around the station all night, Killpath would just have to go out and drag Gun back by the heels once an hour ; ;

If you want a picture get to the corner of Adams and Clark just as fast as you can.

On my way out I told her, `` If you should eh just happen to see your husband, get him to give himself up.

If she runs around with other men, and if you hate her as you say, why not just divorce her ''??

A retired professional killer If he was just a nut, no harm was done.

The next traditional step then was to accept it as the authoritative textbook of the Christian faith just as one would accept a treatise on any earthly `` science '', and I submitted to its conditions according to Christ's invitation and promise that, `` If any man will do his will, he shall know of the doctrine, whether it be of God, or whether I speak of myself '' ( John 7: 17 ).

If you know that he ( God ) is just ( righteous ), know that everyone also who does what is just ( righteous ) has been born of him.

If this is true, then the universe today looks just as it did millions of years ago and as it will look millions of years hence, even though the universe is expanding.

If the voice is just a shade less glorious than it used to be, it is still a beautiful instrument, controlled and flexible.

I kept saying, `` If I could just build up a reputation for myself, make some real money, get to be well known as an illustrator -- like Peter Askington, for instance -- then I could take some time off and paint ''.

Jarry once wrote, expressing some of the bizarre logic of ' pataphysics, " If you let a coin fall and it falls, the next time it is just by an infinite coincidence that it will fall again the same way ; hundreds of other coins on other hands will follow this pattern in an infinitely unimaginable fashion ".

The case of " beauty " is different from mere " agreeableness " because, " If he proclaims something to be beautiful, then he requires the same liking from others ; he then judges not just for himself but for everyone, and speaks of beauty as if it were a property of things.

If the ball has left the field along the sidelines, the referee must decide which team touched the ball last, and award a restart stroke to the opposing team, just like football's throw-in.