Implication Quotes (25 quotes)
… just as the astronomer, the physicist, the geologist, or other student of objective science looks about in the world of sense, so, not metaphorically speaking but literally, the mind of the mathematician goes forth in the universe of logic in quest of the things that are there; exploring the heights and depths for facts—ideas, classes, relationships, implications, and the rest; observing the minute and elusive with the powerful microscope of his Infinitesimal Analysis; observing the elusive and vast with the limitless telescope of his Calculus of the Infinite; making guesses regarding the order and internal harmony of the data observed and collocated; testing the hypotheses, not merely by the complete induction peculiar to mathematics, but, like his colleagues of the outer world, resorting also to experimental tests and incomplete induction; frequently finding it necessary, in view of unforeseen disclosures, to abandon one hopeful hypothesis or to transform it by retrenchment or by enlargement:—thus, in his own domain, matching, point for point, the processes, methods and experience familiar to the devotee of natural science.
[Richard P.] Feynman's cryptic remark, “no one is that much smarter ...,” to me, implies something Feynman kept emphasizing: that the key to his achievements was not anything “magical” but the right attitude, the focus on nature's reality, the focus on asking the right questions, the willingness to try (and to discard) unconventional answers, the sensitive ear for phoniness, self-deception, bombast, and conventional but unproven assumptions.
Given any domain of thought in which the fundamental objective is a knowledge that transcends mere induction or mere empiricism, it seems quite inevitable that its processes should be made to conform closely to the pattern of a system free of ambiguous terms, symbols, operations, deductions; a system whose implications and assumptions are unique and consistent; a system whose logic confounds not the necessary with the sufficient where these are distinct; a system whose materials are abstract elements interpretable as reality or unreality in any forms whatsoever provided only that these forms mirror a thought that is pure. To such a system is universally given the name MATHEMATICS.
God bless all the precious little examples and all their cascading implications; without these gems, these tiny acorns bearing the blueprints of oak trees, essayists would be out of business.
I should have liked to use the word “metaphysics”…, but there are certain words which have accumulated such evil implications that they must either be abandoned, or withdrawn for a period of purification.
If you’re telling a story, it’s very tempting to personalise an animal. To start with, biologists said this fascination with one individual was just television storytelling. But they began to realise that, actually, it was a new way to understand behaviour–following the fortunes of one particular animal could be very revealing and have all kinds of implications in terms of the ecology and general behaviour of the animals in that area.
In our daily lives, we enjoy the pervasive benefits of long-lived robotic spacecraft that provide high-capacity worldwide telecommunications; reconnaissance of Earth’s solid surface and oceans, with far-reaching cultural and environmental implications; much-improved weather and climatic forecasts; improved knowledge about the terrestrial effects of the Sun’s radiations; a revolutionary new global navigational system for all manner of aircraft and many other uses both civil and military; and the science of Earth itself as a sustainable abode of life.
In the expressions we adopt to prescribe physical phenomena we necessarily hover between two extremes. We either have to choose a word which implies more than we can prove, or we have to use vague and general terms which hide the essential point, instead of bringing it out. The history of electrical theories furnishes a good example.
In the summer of 1937, … I told Banach about an expression Johnny [von Neumann] had once used in conversation with me in Princeton before stating some non-Jewish mathematician’s result, “Die Goim haben den folgendenSatzbewiesen” (The goys have proved the following theorem). Banach, who was pure goy, thought it was one of the funniest sayings he had ever heard. He was enchanted by its implication that if the goys could do it, Johnny and I ought to be able to do it better. Johnny did not invent this joke, but he liked it and we started using it.
It is strange that we know so little about the properties of numbers. They are our handiwork, yet they baffle us; we can fathom only a few of their intricacies. Having defined their attributes and prescribed their behavior, we are hard pressed to perceive the implications of our formulas.
Its [science’s] effectiveness is almost inevitable because it narrows the possibility of refutation and failure. Science begins by saying it can only answer this type of question and ends by saying these are the only questions that can be asked. Once the implications and shallowness of this trick are fully realised, science will be humbled and we shall be free to celebrate ourselves once again.
Kuhn … [like Popper] rejects the idea that science grows by accumulation of eternal truths. … But while according to Popper science is “revolution in permanence”, and criticism the heart of the scientific enterprise, according to Kuhn revolution is exceptional and, indeed, extra-scientific, and criticism is, in “normal” times, anathema. … The clash between Popper and Kuhn is not about a mere technical point in epistemology. It concerns our central intellectual values, and has implications not only for theoretical physics but also for the underdeveloped social sciences and even for moral and political philosophy.
Objections … inspired Kronecker and others to attack Weierstrass’ “sequential” definition of irrationals. Nevertheless, right or wrong, Weierstrass and his school made the theory work. The most useful results they obtained have not yet been questioned, at least on the ground of their great utility in mathematical analysis and its implications, by any competent judge in his right mind. This does not mean that objections cannot be well taken: it merely calls attention to the fact that in mathematics, as in everything else, this earth is not yet to be confused with the Kingdom of Heaven, that perfection is a chimaera, and that, in the words of Crelle, we can only hope for closer and closer approximations to mathematical truth—whatever that may be, if anything—precisely as in the Weierstrassian theory of convergent sequences of rationals defining irrationals.
Oersted would never have made his great discovery of the action of galvanic currents on magnets had he stopped in his researches to consider in what manner they could possibly be turned to practical account; and so we would not now be able to boast of the wonders done by the electric telegraphs. Indeed, no great law in Natural Philosophy has ever been discovered for its practical implications, but the instances are innumerable of investigations apparently quite useless in this narrow sense of the word which have led to the most valuable results.
Physicists and astronomers see their own implications in the world being round, but to me it means that only one-third of the world is asleep at any given time and the other two-thirds is up to something.
Psychologism is, I believe, correct only in so far as it insists upon what may be called “methodological individualism” as opposed to “methodological collectivism”; it rightly insists that the “behaviour” and the “actions” of collectives, such as states or social groups, must be reduced to the behaviour and to the actions of human individuals. But the belief that the choice of such an individualist method implies the choice of a psychological method is mistaken.
Pure mathematics … reveals itself as nothing but symbolic or formal logic. It is concerned with implications, not applications. On the other hand, natural science, which is empirical and ultimately dependent upon observation and experiment, and therefore incapable of absolute exactness, cannot become strictly mathematical. The certainty of geometry is thus merely the certainty with which conclusions follow from non-contradictory premises. As to whether these conclusions are true of the material world or not, pure mathematics is indifferent.
Pure Mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form. In addition to these, mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth.
The fact that some geniuses were laughed at does not imply that all who are laughed at are geniuses. They laughed at Columbus, they laughed at Fulton, they laughed at the Wright brothers. But they also laughed at Bozo the clown.
The great mathematician, like the great poet or naturalist or great administrator, is born. My contention shall be that where the mathematic endowment is found, there will usually be found associated with it, as essential implications in it, other endowments in generous measure, and that the appeal of the science is to the whole mind, direct no doubt to the central powers of thought, but indirectly through sympathy of all, rousing, enlarging, developing, emancipating all, so that the faculties of will, of intellect and feeling learn to respond, each in its appropriate order and degree, like the parts of an orchestra to the “urge and ardor” of its leader and lord.
The inherent unpredictability of future scientific developments—the fact that no secure inference can be drawn from one state of science to another—has important implications for the issue of the limits of science. It means that present-day science cannot speak for future science: it is in principle impossible to make any secure inferences from the substance of science at one time about its substance at a significantly different time. The prospect of future scientific revolutions can never be precluded. We cannot say with unblinking confidence what sorts of resources and conceptions the science of the future will or will not use. Given that it is effectively impossible to predict the details of what future science will accomplish, it is no less impossible to predict in detail what future science will not accomplish. We can never confidently put this or that range of issues outside “the limits of science”, because we cannot discern the shape and substance of future science with sufficient clarity to be able to say with any assurance what it can and cannot do. Any attempt to set “limits” to science—any advance specification of what science can and cannot do by way of handling problems and solving questions—is destined to come to grief.
The truly awesome intellectuals in our history have not merely made discoveries; they have woven variegated, but firm, tapestries of comprehensive coverage. The tapestries have various fates: Most burn or unravel in the foot steps of time and the fires of later discovery. But their glory lies in their integrity as unified structures of great complexity and broad implication.
There cannot be design without a designer; contrivance without a contriver; order without choice; arrangement, without any thing capable of arranging; subserviency and relation to a purpose; means suitable to an end, and executing their office in accomplishing that end, without the end ever having been contemplated, or the means accommodated to it. Arrangement, disposition of parts, subserviency of means to an end, relation of instruments to use, imply the preference of intelligence and mind.
These results demonstrate that there is a new polymerase inside the virions of RNA tumour viruses. It is not present in supernatents of normal cells but is present in virions of avian sarcoma and leukemia RNA tumour viruses. The polymerase seems to catalyse the incorporation of deoxyrinonucleotide triphosphates into DNA from an RNA template. Work is being performed to characterize further the reaction and the product. If the present results and Baltimore's results with Rauscher leukemia virus are upheld, they will constitute strong evidence that the DNA proviruses have a DNA genome when they are in virions. This result would have strong implications for theories of viral carcinogenesis and, possibly, for theories of information transfer in other biological systems. [Co-author with American virologist Satoshi Mizutani]
This theme of mutually invisible life at widely differing scales bears an important implication for the ‘culture wars’ that supposedly now envelop our universities and our intellectual discourse in general ... One side of this false dichotomy features the postmodern relativists who argue that all culturally bound modes of perception must be equally valid, and that no factual truth therefore exists. The other side includes the benighted, old-fashioned realists who insist that flies truly have two wings, and that Shakespeare really did mean what he thought he was saying. The principle of scaling provides a resolution for the false parts of this silly dichotomy. Facts are facts and cannot be denied by any rational being. (Often, facts are also not at all easy to determine or specify–but this question raises different issues for another time.) Facts, however, may also be highly scale dependent–and the perceptions of one world may have no validity or expression in the domain of another. The one-page map of Maine cannot recognize the separate boulders of Acadia, but both provide equally valid representations of a factual coastline.