Mathematical Induction Quotes

Mr Baley", said Quemot, "you can't treat human emotions as though they were built about a positronic brain". "I'm not saying you can. Robotics is a deductive science and sociology an inductive one. But mathematics can be made to apply in either case.

Any theory on the principles of mathematics must always be inductive i.e. it must lie in the fact that the theory in question enables us to deduce ordinary mathematics.

The key to our problem lies in mathematical induction. It will be remembered that, in Chapter I., this was the fifth of the five primitive propositions which we laid down about the natural numbers. It stated that any property which belongs to 0, and to the successor of any number which has the property, belongs to all the natural numbers. This was then presented as a principle, but we shall now adopt it as a definition.

the relation of mathematics to the world of temporal change and of phenomenal particularity is direct: less by induction than by what Pierce called abduction – an imaginative jumping off from an open-ended series of particulars.

John Dalton was a very singular Man: He has none of the manners or ways of the world. A tolerable mathematician He gained his livelihood I believe by teaching the mathematics to young people. He pursued science always with mathematical views. He seemed little attentive to the labours of men except when they countenanced or confirmed his own ideas... He was a very disinterested man, seemed to have no ambition beyond that of being thought a good Philosopher. He was a very coarse Experimenter & almost always found the results he required.—Memory & observation were subordinate qualities in his mind. He followed with ardour analogies & inductions & however his claims to originality may admit of question I have no doubt that he was one of the most original philosophers of his time & one of the most ingenious.

Ohm found that the results could be summed up in such a simple law that he who runs may read it, and a schoolboy now can predict what a Faraday then could only guess at roughly. By Ohm's discovery a large part of the domain of electricity became annexed by Coulomb's discovery of the law of inverse squares, and completely annexed by Green's investigations. Poisson attacked the difficult problem of induced magnetisation, and his results, though differently expressed, are still the theory, as a most important first approximation. Ampere brought a multitude of phenomena into theory by his investigations of the mechanical forces between conductors supporting currents and magnets. Then there were the remarkable researches of Faraday, the prince of experimentalists, on electrostatics and electrodynamics and the induction of currents. These were rather long in being brought from the crude experimental state to a compact system, expressing the real essence. Unfortunately, in my opinion, Faraday was not a mathematician. It can scarcely be doubted that had he been one, he would have anticipated much later work. He would, for instance, knowing Ampere's theory, by his own results have readily been led to Neumann's theory, and the connected work of Helmholtz and Thomson. But it is perhaps too much to expect a man to be both the prince of experimentalists and a competent mathematician.

It's all over, it's all over,' I muttered to myself. My grief resembled that of a fainthearted student who has failed an examination: I made a mistake! I made a mistake! Simply because I didn't solve that X, everything was wrong. If only I'd solved that X at the beginning, everything would have been all right. If only I had used deductive methods like everyone else to solve the mathematics of life. To be half-clever was the worst thing I could have done. I alone depended upon the inductive method, and for the simple reason I failed.

But in connection with mathematics the one-sidedness of the Greek genius appears: it reasoned deductively from what appeared self-evident, not inductively from what had been observed. Its amazing successes in the employment of this method misled not only the ancient world, but the greater part of the modern world also. It has only been very slowly that scientific method, which seeks to reach principles inductively from observations of particular facts, has replaced the Hellenic belief in deduction from luminous axioms derived from the mind of the philosopher. For this reason, apart from others, it is a mistake to treat the Greeks with superstitious reverence. Scientific method, though some few among them were the first men who had an inkling of it, is, on the whole, alien to their temper of mind, and the attempt to glorify them by belittling the intellectual progress of the last four centuries has a cramping effect upon modern thought.

The part played by deduction in science is greater than Bacon supposed. Often, when a hypothesis has to be tested, there is a long deductive journey from the hypothesis to some consequence that can be tested by observation. Usually the deduction is mathematical, and in this respect Bacon underestimated the importance of mathematics in scientific investigation. The problem of induction by simple enumeration remains unsolved to this day. Bacon was quite right in rejecting simple enumeration where the details of scientific investigation are concerned, for in dealing with details we may assume general laws on the basis of which, so long as they are taken as valid, more or less cogent methods can be built up. John Stuart Mill framed four canons of inductive method, which can be usefully employed so long as the law of causality is assumed; but this law itself, he had to confess, is to be accepted solely on the basis of induction by simple enumeration. The thing that is achieved by the theoretical organization of science is the collection of all subordinate inductions into a few that are very comprehensive—perhaps only one. Such comprehensive inductions are confirmed by so many instances that it is thought legitimate to accept, as regards them, an induction by simple enumeration. This situation is profoundly unsatisfactory, but neither Bacon nor any of his successors have found a way out of it.

The theory of relativity is a beautiful example of the basic character of the modern development of theory. That is to say, the hypotheses from which one starts become ever more abstract and more remote from experience. But in return one comes closer to the preeminent goal of science, that of encompassing a maximum of empirical contents through logical deduction with a minimum of hypotheses or axioms. The intellectual path from the axioms to the empirical contents or to the testable consequences becomes, thereby, ever longer and more subtle. The theoretician is forced, ever more, to allow himself to be directed by purely mathematical, formal points of view in the search for theories, because the physical experience of the experimenter is not capable of leading us up to the regions of the highest abstraction. Tentative deduction takes the place of the predominantly inductive methods appropriate to the youthful state of science. Such a theoretical structure must be quite thoroughly elaborated in order for it to lead to consequences that can be compared with experience. It is certainly the case that here, as well, the empirical fact is the all-powerful judge. But its judgment can be handed down only on the basis of great and difficult intellectual effort that first bridges the wide space between the axioms and the testable consequences. The theorist must accomplish this Herculean task with the clear understanding that this effort may only be destined to prepare the way for a death sentence for his theory. One should not reproach the theorist who undertakes such a task by calling him a fantast; instead, one must allow him his fantasizing, since for him there is no other way to his goal whatsoever. Indeed, it is no planless fantasizing, but rather a search for the logically simplest possibilities and their consequences.

Chesterton's topic is nothing less than the fundamental contrast between deductive logic, true of all possible worlds, and inductive logic, capable only of telling us how we may reasonably expect this world to behave. Let us hasten to add that Chesterton's analysis is in full agreement with the views of modern logicians. Perhaps his "test of the imagination" is not strictly accurate--who can "imagine" the four-dimensional constructions of relativity?-but in essence his position is unassailable. Logical and mathematical statements are true by definition. They are "empty tautologies," to use a current phrase, like the impressive maxim that there are always six eggs in half a dozen. Nature, on the other hand, is under no similar constraints. Fortunately, her "weird repetitions," as GK calls them, often conform to surprisingly low-order equations. But as Hume and others before Hume made clear, there is no logical reason why she should behave so politely.

The admissibility of complete induction cannot only not be proved, but it ought neither to be considered as a separate axiom nor as a separately seen intuitive truth. Complete induction is an act of mathematical constructing, which is already justified by the basic intuition of mathematics.

We have then many kinds of intuition; first, the appeal to the senses and the imagination; next, generalization by induction, copied, so to speak, from the procedures of the experimental sciences; finally, we have the intuition of pure number, whence arose the second of the axioms just enunciated, which is able to create the real mathematical reasoning.

Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs.

Hilbert’s logic is a hollow structure, built up from differently coloured types of bricks, in which the arithmetic of the natural numbers is tacitly assumed, including induction; but it cannot prove anything which is in some vague way connected to our known mathematical systems.

Hilbert does not build on the foundation of logic and arithmetic, but on the system of signs thereof; in the construction he uses the logic (syllogism from a general theorem for x) and the arithmetic (mathematical induction) as something meaningless and independent. Moreover he presupposes as known the whole of mathematics as a guideline in the introduction of new symbols; and he employs subtle logical reasonings to convince us that he is on the right track.

What is at issue here is not the familiar construct of formal mathematics, but a belief in the existence of ω (the set of natural numbers) prior to all mathematical constructions. What is the origin of this belief? The famous saying by Kronecker that God created the numbers, all the rest is the work of Man, presumably was not meant to be taken seriously. Nowhere in the Book of Genesis do we find the passage: And God said, let there be numbers, and there were numbers; odd and even created he them, and he said unto them be fruitful and multiply; and he commanded them to keep the laws of induction. No, the belief in ω stems from the speculations of Greek philosophy on the existence of ideal entities or the speculations of German philosophy on a priori categories of thought.

The process of objectifying the world through the primordial intuition of "repetition in time" and "following in time" gains in generality by the construction of mathematics from the same primordial intuition, without reference to direct applicability. In this way man has a ready-made supply of unreal causal sequences at his disposal, just waiting for an opportunity to be projected into reality. One should bear in mind that in mathematical systems with no time coordinate, all relations in practical applications clearly become causal relations in time; e.g. Euclidean geometry when applied to reality shows a causal connection between the results of different measurements made by means of the group of rigid bodies. Needless to say, in the application of a mathematical system, in general, only a fraction of the elements and substructures finds their correspondence in reality; the remainder plays the role of and unreal "physical hypothesis." Similarly, even with a limited development of method, the observed sequences no longer consist exclusively of phenomena evoked by man himself (acts without any direct instinctive aim, but carried out solely to complete the causal system into a more manageable one). The simplest example is the sound image (or written symbol) of number as a result of counting, or the sound image (or written symbol) of number as a result of measuring (this example shows how infinitely many causal sequences can be brought together under the viewpoint of one single law of causality on the basis of a mapping the numbers through mathematical induction.)

Why then does this judgement [mathematical induction] force itself upon us with an irresistible evidence? It is because it is only the affirmation of the power of the mind which knows itself capable of conceiving the indefinite repetition of the same act when once this act is possible. The mind has a direct intuition of this power, and experience can only give occasion for using it and thereby becoming conscious of it.

same phenomenon and concludes that the phenomenon will always occur. Conclusions obtained by induction seem well warranted

If it seems unfair for banks to discriminate against you because your high school buddy is bad at paying his bills or because you like something that a lot of loan defaulters also like, well, it is. And it points to a basic problem with induction, the logical method by which algorithms use data to make predictions. Philosophers have been wrestling with this problem since long before there were computers to induce with. While you can prove the truth of a mathematical proof by arguing it out from first principles, the philosopher David Hume pointed out in 1772 that reality doesn’t work that way. As the investment cliché has it, past performance is not indicative of future results.

This chart contrasts predictive and prospective thinking: Predictive Thinking Prospective Thinking Mindset Forecasting, “We expect …” Preparing, “But what if …” Goal Reduce or even discard uncertainty, fight ambiguity Live with uncertainty, embrace ambiguity, plan for set of contingencies Level of uncertainty Average High Method Extrapolating from present and past Open, imaginative Approach Categorical, assumes continuity Global, systemic, anticipates disruptive events Information inputs Quantitative, objective, known Qualitative (whether quantifiable or not), subjective, known or unknown Relationships Static, stable structures Dynamic, evolving structures Technique Established quantitative models (economics, mathematics, data) Developing scenarios using qualitative approaches (often building on megatrends) Evaluation method Numbers Criteria Attitude toward the future Passive or reactive (the future will be) Proactive and creative (we create or shape the future) Way of thinking Generally deduction Greater use of induction

Mr. Rothschild discovered the missing passive component of economic theory known as economic inductance. He, of course, did not think of his discovery in these 20th-century terms, and, to be sure, mathematical analysis had to wait for the Second Industrial Revolution,

In developing formal logic, Aristotle took Greek mathematics as his model. Like his predecessors Socrates and Plato, Aristotle was impressed with the rigor and precision of geometrical proofs. His goal was to formalize and generalize those proof procedures and apply them to philosophy, science, and all other branches of knowledge. Yet not all subjects are equally amenable to formalization. Greek mathematics achieved its greatest successes in astronomy, where Ptolemy's calculations remained the standard of precision for centuries. But other subjects, such as medicine and law, depend more on deep experience than on brilliant mathematical calculations. Significantly, two of the most penetrating criticisms of logic were written by the physician Sextus Empiricus in the second century AD and by the legal scholar Ibn Taymiyya in the fourteenth century. Sextus Empiricus, as his nickname suggests, was an empiricist. By profession, he was a physician; philosophically, he was an adherent of the school known as the Skeptics. Sextus maintained that all knowledge must come from experience. As an example, he cited the following syllogism: Every human is an animal. Socrates is human. Therefore, Socrates is an animal. Sextus admitted that this syllogism represents a valid inference pattern, but he questioned the source of evidence for the major premise Every human is an animal. A universal proposition that purports to cover every instance of some category must be derived by induction from particulars. If the induction is incomplete, then the universal proposition is not certain, and there might be some human who is not an animal. But if the induction is complete, then the particular instance Socrates must have been examimed already, and the syllogism is redundant or circular. Since every one of Aristotle's valid forms of syllogisms contains at least one universal affirmative or universal negative premise, the same criticisms apply to all of them: the conclusion must be either uncertain or circular. The Aristotelians answered Sextus by claiming that universal propositions may be true by definition: since the type Human is defined as rational animal, the essence of human includes animal; therefore, no instance of human that was not an animal could exist. This line of defense was attacked by the Islamic jurist and legal scholar Taqi al-Din Ibn Taymiyya. Like Sextus, Ibn Taymiyya agreed that the form of a syllogism is valid, but he did not accept Aristotle's distinction between essence and accident (Hallaq 1993). According to Aristotle, the essence of human includes both rational and animal. Other attributes, such as laughing or being a featherless biped, might be unique to humans, but they are accidental attributes that could be different without changing the essence. Ibn Taymiyya, however, maintained that the distinction between essence and accident was arbitrary. Human might just as well be defined as laughing animal, with rational as an accidental attribute.