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Good mathematicians see analogies. Great mathematicians see analogies between analogies.
Sep 10, 2025
I think it's important to reason from first principles rather than by analogy. The normal way we conduct our lives is we reason by analogy. [With analogy] we are doing this because it's like something else that was done, or it is like what other people are doing. [With first principles] you boil things down to the most fundamental truths…and then reason up from there.
A mathematician's work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks.
Don’t just follow the trend. You may have heard me say that it’s good to think in terms of the physics approach of first principles. Which is, rather than reasoning by analogy, you boil things down to the most fundamental truths you can imagine and you reason up from there.
I trust that a graduate student some day will write a doctoral essay on the influence of the Munich analogy on the subsequent history of the twentieth century. Perhaps in the end he will conclude that the multitude of errors committed in the name of Munich may exceed the original error of 1938.
All penguins are the same below the surface, which I think is as perfect an analogy as we're likely to get for the futility of racism.
All perception of truth is the detection of an analogy.
Music and politics are in essence about communication. Without over-stretching the analogy I do feel a sense of rhythm is important in getting your message across.
After all, the past is our only real guide to the future, and historical analogies are instruments for distilling and organizing the past and converting it to a map by which we can navigate.
While analogies are useful, however, they can also be misleading. They smuggle in assumptions that can be wrong.
Analogies, it is true, decide nothing, but they can make one feel more at home.
I see the origin of the irresistible attraction of metaphor and analogy, the explanation of our strange and permanent need to find similarities in things. I can scarcely refrain from suspecting some ancient, diffused magnetism; a call from the center of things; a dim, almost lost memory, or perhaps a presentiment, pointless in so puny a being, of a universal syntax.
In early times every sort of advantage tends to become a military advantage; such is the best way, then, to keep it alive. But the Jewish advantage never did so; beginning in religion, contrary to a thousand analogies, it remained religious.
By bringing together what we know and what we don't know through analogy, metaphorical thinking strikes the spark that ignites discovery.
We often attribute 'understanding' and other cognitive predicates by metaphor and analogy to cars, adding machines, and other artifacts, but nothing is proved by such attributions.
One of the biggest mistakes that people make when they think about memes is they try to extend on the analogy with genes. That's not how it works. It works by realizing the concept of a replicator.
I take the walk to be the externalization of an interior seeking so that the analogy is first of all between the external and the internal.
The principle of equity first came into evidence in Roman jurisprudence and was derived by analogy from the physical meaning of the word.
Beanie and Cecil was the first cartoon I remember watching and I think there are analogies.
Life is like an analogy.
We know that attention acts as a lightning rod. Merely by concentrating on something one causes endless analogies to collect around it, even penetrate the boundaries of the subject itself: an experience that we call coincidence, serendipity – the terminology is extensive. My experience has been that in these circular travels what is really significant surrounds a central absence, an absence that, paradoxically, is the text being written or to be written.
If one looks at the different problems of the integral calculus which arise naturally when one wishes to go deep into the different parts of physics, it is impossible not to be struck by the analogies existing.
Many people tend to look at programming styles and languages like religions: if you belong to one, you cannot belong to others. But this analogy is another fallacy.
Mathematical analysis is as extensive as nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures:;; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind. It's chief attribute is clearness; it has no marks to express confused notations. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them.
Science is nothing but the finding of analogy, identity, in the most remote parts.
We know nothing of what will happen in future, but by the analogy of experience.
Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.
Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements.
Imagination is an almost divine faculty which, without recourse to any philosophical method, immediately perceives everything: the secret and intimate connections between things, correspondences and analogies.
The human mind delights in finding pattern—so much so that we often mistake coincidence or forced analogy for profound meaning. No other habit of thought lies so deeply within the soul of a small creature trying to make sense of a complex world not constructed for it.
The world is full of hopeful analogies and handsome, dubious eggs, called possibilities.
If genius has any common denominator, I would propose breadth of interest and the ability to construct fruitful analogies between fields.
We have said that consciousness is an operation rather than a thing, a repository, or a function. It operates by way of analogy, by way of constructing an analog space with an analog 'I' that can observe that space, and move metaphorically in it. It operates on any reactivity, excerpts relevant aspects, narratizes and conciliates them together in a metaphorical space where such meanings can be manipulated like things in space.
The eye instinctively looks for analogies and amplifies them, so that a face imagined in the pattern of a wallpaper may become more vivid than a photograph.
The whole of natural theologyresolves itself into one simple, though somewhat ambiguous proposition, That the cause or causesof order in the universe probably bear some remote analogy to human intelligence.
Sin2 φ is odious to me, even though Laplace made use of it; should it be feared that sin2 φ might become ambiguous, which would perhaps never occur, or at most very rarely when speaking of sin(φ2), well then, let us write (sin φ)2, but not sin2 φ, which by analogy should signify sin (sin φ)
One of the things that I hope will distinguish Amazon.com is that we continue to be a company that defies easy analogy. This requires a lot of innovation, and innovation requires a lot of random walk.
Can you not see the numerous designs made by God as signs, similitudes, or analogies of resurrection? He has placed them in every era, the alteration of day and night, even in the coming and going of clouds.
Turbulence is a condition that we all experience during a flight when the plane is bouncing around by competing air currents. By analogy, the economy may bounce around a lot because of competing currents of public moods and investments. One week everyone might be optimistic and then suddenly something happens to turn everyone into pessimists. Investment dries up and investors become risk averse. A sudden piece of good news then turns around the public mood.
Nowadays a lot of what was wrong with me would no doubt be ascribed to Attention Deficit Disorder, tartrazine food colouring, dairy produce and air pollution. A few hundred years earlier it would have been demons, still the best analogy I think, but not much help when it comes to a cure.
We use the Air Force analogy: there were expensive things they had to do to get a cockpit suitable for a lot of pilots, like wraparound windshields, but their initial solutions, when they realized average didn't work, were adjustable seats. How in the world did they not already have adjustable seats in their planes? We're looking for adjustable seats for education, for basic things that we can do.
If I may make a football analogy, we're a team whether we're a football team or community or the United States of America. We are part of a team and I believe the people on that team have a right, but they also have the obligation if there is something that is not good or we don't agree on, to speak about it.
The analogies between science and art are very good as long as you are talking about the creation and the performance. The creation is certainly very analogous. The aesthetic pleasure of the craftsmanship of performance is also very strong in science.
Analogies have tied things together for me, personally. The fundamental one for me is the analogy between your relationship to your spouse and your relationship to your place. Both need to be a settled commitment and both involve continuous learning and adjusting.
The analogy between the mind and a computer fails for many reasons. The brain is constructed by principles that assure diversity and degeneracy. Unlike a computer, it has no replicative memory. It is historical and value driven. It forms categories by internal criteria and by constraints acting at many scales, not by means of a syntactically constructed program. The world with which the brain interacts is not unequivocally made up of classical categories.
I think that metaphor really is a key to explaining thought and language. The human mind comes equipped with an ability to penetrate the cladding of sensory appearance and discern the abstract construction underneath - not always on demand, and not infallibly, but often enough and insightfully enough to shape the human condition. Our powers of analogy allow us to apply ancient neural structures to newfound subject matter, to discover hidden laws and systems in nature, and not least, to amplify the expressive power of language itself.
Joy Rains has a gift. A gift for explaining abstract concepts like meditation through clever analogies and metaphors. FOR THE FIRST TIME in my hectic, harried life I actually UNDERSTAND how meditation is supposed to work and WHY I should give it a TRY!
Painting is the making of an analogy for something non-visual and incomprehensible - giving it form and bringing it within reach. And that is why good paintings are incomprehensible. Creating the incomprehensible has absolutely nothing to do with turning out any old bunkum, because bunkum is always comprehensible.
The formation of the life of a person in the womb is the work of God, and it is not merely a mechanical process but a work on the analogy of weaving or knitting: "Thou didst knit me together in my mother's womb" (psalm 139:13). The life of the unborn is the knitting of God, and what He is knitting is a human being in His own image, unlike any other creature in the universe... The destruction of conceived human life - whether embryonic, fetal, or viable - is an assault on the unique person-forming work of God.
So in this case, Kalamas, don't go by reports, by legends, by traditions, by scripture, by logical conjecture, by inference, by analogies, by agreement through pondering views, by probability, or by the thought, 'This contemplative is our teacher.' When you know for yourselves that, 'These qualities are unskillful; these qualities are blameworthy; these qualities are criticized by the wise; these qualities, when adopted & carried out, lead to harm & to suffering' - then you should abandon them.