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Home > Dictionary of Science Quotations > Scientist Names Index W > Alfred North Whitehead Quotes > Mathematics

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Alfred North Whitehead
(15 Feb 1861 - 30 Dec 1947)

English mathematician and philosopher who worked in logic, physics, and later in his life spent more time on the philosophy of science and metaphysics. He worked with Bertrand Russell on Principia Mathematica which shows that logic underlies all mathematics.



All science as it grows toward perfection becomes mathematical in its ideas.
— Alfred North Whitehead
In An Introduction to Mathematics (1911), 14. This is part of a longer quote that begins, “In modern times the belief that the ultimate explanation…”, on the Alfred North Whitehead Quotes page of this website.
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Archimedes, who combined a genius for mathematics with a physical insight, must rank with Newton, who lived nearly two thousand years later, as one of the founders of mathematical physics. … The day (when having discovered his famous principle of hydrostatics he ran through the streets shouting Eureka! Eureka!) ought to be celebrated as the birthday of mathematical physics; the science came of age when Newton sat in his orchard.
— Alfred North Whitehead
In An Introduction to Mathematics (1911), 37.
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Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world could have more astonished a Greek mathematician than to learn that, under the influence of compulsory education, the whole population of Western Europe, from the highest to the lowest, could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility. … Our modern power of easy reckoning with decimal fractions is the most miraculous result of a perfect notation.
— Alfred North Whitehead
In Introduction to Mathematics (1911), 59.
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Definition of Mathematics.—It has now become apparent that the traditional field of mathematics in the province of discrete and continuous number can only be separated from the general abstract theory of classes and relations by a wavering and indeterminate line. Of course a discussion as to the mere application of a word easily degenerates into the most fruitless logomachy. It is open to any one to use any word in any sense. But on the assumption that “mathematics” is to denote a science well marked out by its subject matter and its methods from other topics of thought, and that at least it is to include all topics habitually assigned to it, there is now no option but to employ “mathematics” in the general sense of the “science concerned with the logical deduction of consequences from the general premisses of all reasoning.”
— Alfred North Whitehead
In article 'Mathematics', Encyclopedia Britannica (1911, 11th ed.), Vol. 17, 880. In the 2006 DVD edition of the encyclopedia, the definition of mathematics is given as “The science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects.” [Premiss is a variant form of “premise”. —Webmaster]
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I will not go so far as to say that to construct a history of thought without profound study of the mathematical ideas of successive epochs is like omitting Hamlet from the play which is named after him. That would be claiming too much. But it is certainly analogous to cutting out the part of Ophelia. This simile is singularly exact. For Ophelia is quite essential to the play, she is very charming-and a little mad. Let us grant that the pursuit of mathematics is a divine madness of the human spirit, a refuge from the goading urgency of contingent happenings.
— Alfred North Whitehead
In Science and the Modern World (1926), 31.
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In modern times the belief that the ultimate explanation of all things was to be found in Newtonian mechanics was an adumbration of the truth that all science, as it grows towards perfection, becomes mathematical in its ideas.
— Alfred North Whitehead
In An Introduction to Mathematics (1911), 13-14. [To suggest, disclose, or outline partially, produces an “adumbration”, which gives only the main facts and not the details. —Webmaster]
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It is a safe rule to apply that, when a mathematical or philosophical author writes with a misty profoundity, he is talking nonsense.
— Alfred North Whitehead
In An Introduction to Mathematics (1911), 227.
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Mathematical reasoning is deductive in the sense that it is based upon definitions which, as far as the validity of the reasoning is concerned (apart from any existential import), needs only the test of self-consistency. Thus no external verification of definitions is required in mathematics, as long as it is considered merely as mathematics.
— Alfred North Whitehead
In Universal Algebra (1898), Preface, vi.
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Mathematics as a science commenced when first someone, probably a Greek, proved propositions about any things or about some things, without specification of definite particular things. These propositions were first enunciated by the Greeks for geometry; and, accordingly, geometry was the great Greek mathematical science.
— Alfred North Whitehead
In An Introduction to Mathematics (1911), 15.
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Mathematics in its widest signification is the development of all types of formal, necessary, deductive reasoning.
— Alfred North Whitehead
In Universal Algebra (1898), Preface, vi.
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Mathematics is often considered a difficult and mysterious science, because of the numerous symbols which it employs. Of course, nothing is more incomprehensible than a symbolism which we do not understand. … But this is not because they are difficult in themselves. On the contrary they have invariably been introduced to make things easy. … [T]he symbolism is invariably an immense simplification. It … represents an analysis of the ideas of the subject and an almost pictorial representation of their relations to each other.
— Alfred North Whitehead
In Introduction to Mathematics (1911), 59-60.
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Mathematics … is necessarily the foundation of exact thought as applied to natural phenomena.
— Alfred North Whitehead
In An Introduction to Mathematics (1911), 8.
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MATHEMATICS … the general term for the various applications of mathematical thought, the traditional field of which is number and quantity. It has been usual to define mathematics as “the science of discrete and continuous magnitude.”
— Alfred North Whitehead
Opening statement in article 'Mathematics', Encyclopedia Britannica (1911, 11th ed.), Vol. 17, 878. Whitehead then indicated this was an inadequate definition, which he then discussed at length and tried to give an improved definition later in the article. See the quote beginning “Definition of Mathematics…” on the Alfred North Whitehead Quotes page on this website.
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No part of Mathematics suffers more from the triviality of its initial presentation to beginners than the great subject of series. Two minor examples of series, namely arithmetic and geometric series, are considered; these examples are important because they are the simplest examples of an important general theory. But the general ideas are never disclosed; and thus the examples, which exemplify nothing, are reduced to silly trivialities.
— Alfred North Whitehead
In An Introduction to Mathematics (1911), 194.
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The anxious precision of modern mathematics is necessary for accuracy, … it is necessary for research. It makes for clearness of thought and for fertility in trying new combinations of ideas. When the initial statements are vague and slipshod, at every subsequent stage of thought, common sense has to step in to limit applications and to explain meanings. Now in creative thought common sense is a bad master. Its sole criterion for judgment is that the new ideas shall look like the old ones, in other words it can only act by suppressing originality.
— Alfred North Whitehead
In Introduction to Mathematics (1911), 157.
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The first acquaintance which most people have with mathematics is through arithmetic. That two and two make four is usually taken as the type of a simple mathematical proposition which everyone will have heard of. … The first noticeable fact about arithmetic is that it applies to everything, to tastes and to sounds, to apples and to angels, to the ideas of the mind and to the bones of the body.
— Alfred North Whitehead
In An Introduction to Mathematics (1911), 9.
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The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every province of thought, or of external experience, in which the succession of thoughts, or of events can be definitely ascertained and precisely stated. So that all serious thought which is not philosophy, or inductive reasoning, or imaginative literature, shall be mathematics developed by means of a calculus.
— Alfred North Whitehead
In Universal Algebra (1898), Preface.
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The invention of the differential calculus marks a crisis in the history of mathematics. The progress of science is divided between periods characterized by a slow accumulation of ideas and periods, when, owing to the new material for thought thus patiently collected, some genius by the invention of a new method or a new point of view, suddenly transforms the whole subject on to a higher level.
— Alfred North Whitehead
In An Introduction to Mathematics (1911), 217. Whitehead continued by quoting the poet, Percy Shelley, who compared the slow accumulation of thoughts leading to an avalanche following the laying down of a great truth. See the poetic quote beginning, “The sun-awakened avalanche…” on the Percy Shelley Quotations page.
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The originality of mathematics consists in the fact that in mathematical science connections between things are exhibited which, apart from the agency of human reason, are extremely unobvious.
— Alfred North Whitehead
In Science and the Modern World (1938), 32.
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The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours. So long as you are dealing with pure mathematics, you are in the realm of complete and absolute abstraction. … Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about.
— Alfred North Whitehead
In Science and the Modern World: Lowell Lectures, 1925 (1925), 31.
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The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours. … Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about.
— Alfred North Whitehead
In 'Mathematics', Science and the Modern World (1926, 2011), 27.
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The science of pure mathematics … may claim to be the most original creation of the human spirit.
— Alfred North Whitehead
In Science and the Modern World: Lowell Lectures, 1925 (1925), 29.
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The science [of mathematics] has grown to such vast proportion that probably no living mathematician can claim to have achieved its mastery as a whole.
— Alfred North Whitehead
In An Introduction to Mathematics (1911), 262.
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The study of mathematics is apt to commence in disappointment. The important applications of the science, the theoretical interest of its ideas, and the logical rigour of its methods all generate the expectation of a speedy introduction to processes of interest. We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it.
— Alfred North Whitehead
Opening to An Introduction to Mathematics (1911), 7.
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The study of mathematics is apt to commence in disappointment. … We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this greatest science eludes the efforts of our mental weapons to grasp it.
— Alfred North Whitehead
Opening of Chap 1, in An Introduction to Mathematics (1911), 7.
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The whole of Mathematics consists in the organization of a series of aids to the imagination in the process of reasoning.
— Alfred North Whitehead
In Universal Algebra (1898), 12.
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There is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain.
— Alfred North Whitehead
In An Introduction to Mathematics (1911), 27.
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Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
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