Robert Langlands quote: “Mathematical maturity is anyhow an uncertain concept, for the mind’s natural competence seems to change with age, its p…”

“Among the various paradigmatic changes in science and mathematics in this century, one such change concerns the concept of uncertainty.”

George Klir (1932–2016) American computer scientist

In science, this change has been manifested by a gradual transition from the traditional view, which insists that uncertainty is undesirable in science and should be avoided by all possible means, to an alternative view, which is tolerant of uncertainty and insists that science cannot avoid it. According to the traditional view, science should strive for certainty in all its manifestations (precision, specificity, sharpness, consistency, etc.); hence, uncertainty (imprecision, nonspecificity, vagueness, inconsistency,etc.) is regarded as unscientific. According to the alternative (or modem) view, uncertainty is considered essential to science; it is not only an unavoidable plague, but it has, in fact, a great utility.
Source: Fuzzy sets and fuzzy logic (1995), p. 1.

“The eagle must age, renew itself, change its beak and remove its flying feathers. This is a natural process for humans.”

Cristina Lizardo (1959) Dominican Republic academic

Source: Cristina Lizardo: “It’s time to show what is being done to damage the image of the PLD.” https://www.catholictranscript.org/cristina-lizardo-its-time-to-show-what-is-being-done-to-damage-the-image-of-the-pld/ (27 June 2021)

“In order for there to be a variable quantity in some mathematical study, the domain of its variability must strictly speaking be known beforehand through a definition. However, this domain cannot itself be something variable, since otherwise each fixed support for the study would collapse. Thus this domain is a definite, actually infinite set of values. Hence each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite.”

Georg Cantor (1845–1918) mathematician, inventor of set theory

"Über die verschiedenen Ansichten in Bezug auf die actualunendlichen Zahlen" ["Over the different views with regard to the actual infinite numbers"] - Bihand Till Koniglen Svenska Vetenskaps Akademiens Handigar (1886)

“Classical mechanics is mathematical not only in the sense that it makes use of mathematical terms and methods for abbreviating arguments which might, if necessary, also be expressed in the language of everyday speech; it is so also in the much more stringent sense that its basic concepts are mathematical concepts, that mechanics itself is a mathematics.”

Eduard Jan Dijksterhuis (1892–1965) Dutch historian

Source: The mechanization of the world picture, 1961, p. 499

“Naturally moving water augments itself. It improves its quality and matures considerably. Its boiling and freezing points change, and wise Nature makes use of this phenomenon to raise water, without using pumping equipment, to the highest mountain peaks, to appear as mountain springs. This conception of raising water is not to be taken literally, since in this context it is concerned with the natural process of propagation and purification. This in turn helps towards the expansion of air by creating an air cover, which serves to develop a higher form of life.”

Viktor Schauberger (1885–1958) austrian philosopher and inventor

Living Water

“To exist is to change, to change is to mature, to mature is to go on creating oneself endlessly.”

Henri Bergson (1859–1941) French philosopher

“It is known that the mathematics prescribed for the high school [Gymnasien] is essentially Euclidean, while it is modern mathematics, the theory of functions and the infinitesimal calculus, which has secured for us an insight into the mechanism and laws of nature. Euclidean mathematics is indeed, a prerequisite for the theory of functions, but just as one, though he has learned the inflections of Latin nouns and verbs, will not thereby be enabled to read a Latin author much less to appreciate the beauties of a Horace, so Euclidean mathematics, that is the mathematics of the high school, is unable to unlock nature and her laws. Euclidean mathematics assumes the completeness and invariability of mathematical forms; these forms it describes with appropriate accuracy and enumerates their inherent and related properties with perfect clearness, order, and completeness, that is, Euclidean mathematics operates on forms after the manner that anatomy operates on the dead body and its members.
On the other hand, the mathematics of variable magnitudes—function theory or analysis—considers mathematical forms in their genesis. By writing the equation of the parabola, we express its law of generation, the law according to which the variable point moves. The path, produced before the eyes of the 113 student by a point moving in accordance to this law, is the parabola.
If, then, Euclidean mathematics treats space and number forms after the manner in which anatomy treats the dead body, modern mathematics deals, as it were, with the living body, with growing and changing forms, and thus furnishes an insight, not only into nature as she is and appears, but also into nature as she generates and creates,—reveals her transition steps and in so doing creates a mind for and understanding of the laws of becoming. Thus modern mathematics bears the same relation to Euclidean mathematics that physiology or biology … bears to anatomy. But it is exactly in this respect that our view of nature is so far above that of the ancients; that we no longer look on nature as a quiescent complete whole, which compels admiration by its sublimity and wealth of forms, but that we conceive of her as a vigorous growing organism, unfolding according to definite, as delicate as far-reaching, laws; that we are able to lay hold of the permanent amidst the transitory, of law amidst fleeting phenomena, and to be able to give these their simplest and truest expression through the mathematical formulas”

Christian Heinrich von Dillmann (1829–1899) German educationist

Source: Die Mathematik die Fackelträgerin einer neuen Zeit (Stuttgart, 1889), p. 37.

“To my mind, the expression of divinity is in variety, and the more variable the creation, the more variable the creatures that surround us, botanical and zoological, the more chance we have to learn and to see into life itself, nature itself.”

Sheri S. Tepper (1929–2016) American fiction writer

Locus interview (1998)
Context: To my mind, the expression of divinity is in variety, and the more variable the creation, the more variable the creatures that surround us, botanical and zoological, the more chance we have to learn and to see into life itself, nature itself. If we were just human beings, living in a spaceship, with an algae farm to give us food, we would not be moved to learn nearly as many things as we are moved by living on a world, surrounded by all kinds of variety. And when I see that variety being first decimated, and then halved — and I imagine in another hundred years it may be down by 90% and there'll be only 10% of what we had when I was a child — that makes me very sad, and very despairing, because we need variety. We came from that, we were born from that, it's our world, the world in which we became what we have become.

“I maintain that in every special natural doctrine only so much science proper is to be met with as mathematics; for… science proper, especially of nature, requires a pure portion, lying at the foundation of the empirical, and based upon à priori knowledge of natural things. …the conception should be constructed. But the cognition of the reason through construction of conceptions is mathematical. A pure philosophy of nature in general, namely, one that only investigates what constitutes a nature in general, may thus be possible without mathematics; but a pure doctrine of nature respecting determinate natural things (corporeal doctrine and mental doctrine), is only possible by means of mathematics; and as in every natural doctrine only so much science proper is to be met with therein as there is cognition à priori, a doctrine of nature can only contain so much science proper as there is in it of applied mathematics.”

Immanuel Kant book Metaphysical Foundations of Natural Science

Preface, Tr. https://books.google.com/books?id=OCJLAAAAMAAJ Ernest Belfort Bax (1883)
Metaphysical Foundations of Natural Science (1786)

“A linguistic variable is a variable whose values are words or sentences in a natural or synthetic language.”

Lotfi A. Zadeh (1921–2017) Electrical engineer and computer scientist

Variant: A linguistic variable is defined as a variable whose values are sentences in a natural or artificial language.
Source: 1970s, Outline of a new approach to the analysis of complex systems and decision processes (1973), p. 28

“Mathematical maturity is anyhow an uncertain concept, for the mind’s natural competence seems to change with age, its purview variable.”

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