### The Mathematical Sciences in 2025

### I asked: “What does mathematics mean to you?” And some people answered: “The manipulation of numbers, the manipulation of structures.” And if I had asked what music means to you, would you have answered: “The manipulation of notes?”

– S. Lang, The Beauty of Doing Mathematics

### The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful.

– J. H. Poincare

### Professor Robert H. Lewis writes: “Mathematics is not about answers, it’s about processes.” He observes that, far too often, the way in which students learn math is equivalent to building a scaffold without ever constructing the building that the scaffolding is intended to support. “The real ‘building’ in the mathematics sense is the true mathematical understanding, the true ability to think, perceive, and analyze mathematically.”

What is math? Dr. David Moursund

What is math? Sanderson M. Smith

Dr. Robert Lewis writes:

### The great misconception about mathematics — and it stifles and thwarts more students than any other single thing — is the notion that mathematics is about formulas and cranking out computations. It is the unconsciously held delusion that mathematics is a set of rules and formulas that have been worked out by God knows who for God knows why, and the student’s duty is to memorize all this stuff. Such students seem to feel that sometime in the future their boss will walk into the office and demand “Quick, what’s the quadratic formula?” Or, “Hurry, I need to know the derivative of 3x^2 – 6x + 7.” There are no such employers.

### What is mathematics really like? Mathematics is not about answers, it’s about processes. Let me give a series of parables to try to get to the root of the misconceptions and to try to illuminate what mathematics IS all about. None of these analogies is perfect, but all provide insight.

### When a new building is made, a skeleton of steel struts called the scaffolding is put up first. The workers walk on the scaffolding and use it to hold equipment as they begin the real task of constructing the building. The scaffolding has no use by itself. It would be absurd to just build the scaffolding and then walk away, thinking that something of value has been accomplished.

### Yet this is what seems to occur in all too many mathematics classes in high schools. Students learn formulas and how to plug into them. They learn mechanical techniques for solving certain equations or taking derivatives. But all of these things are just the scaffolding. They are necessary and useful, sure, but by themselves they are useless. Doing only the superficial and then thinking that something important has happened is like building only the scaffolding.

### The real “building” in the mathematics sense is the true mathematical understanding, the true ability to think, perceive, and analyze mathematically.

### Ready for the Big Play: Professional athletes spend hours in gyms working out on equipment of all sorts. Special trainers are hired to advise them on workout schedules. They spend hours running on treadmills. Why do they do that? Are they learning skills necessary for playing their sport, say basketball?

### Imagine there are three seconds left in the seventh game of the NBA championship. The score is tied. Time out. The pressure is intense. The coach is huddling with his star players. He says to one, “OK Michael, this is it. You know what to do.” And Michael says, “Right coach. Bring in my treadmill!”

### Duh! Of course not! But then what was all that treadmill time for? If the treadmill is not seen during the actual game, was it just a waste to use it? Were all those trainers wasting their time? Of course not. It produced (if it was done right!) something of value, namely stamina and aerobic capacity. Those capacities are of enormous value even if they cannot be seen in any immediate sense. So too does mathematics education produce something of value, true mental capacity and the ability to think.

### Dr. Robert H. Lewis, Fordham University

What is Mathematics? The Most Misunderstood Subject

## Is math made by humans or a feature of the universe?

Aei ho theos geōmetreî. “God always geometrizes” — Plato

Join NOVA on a mathematical mystery tour—a provocative exploration of math’s astonishing power across the centuries. We discover math’s signature in the swirl of a nautilus shell, the whirlpool of a galaxy, and the spiral in the center of a sunflower. Math was essential to everything from the first wireless radio transmissions to the prediction and discovery of the Higgs boson and the successful landing of rovers on Mars. Astrophysicist and writer Mario Livio, along with a colorful cast of mathematicians, physicists, and engineers, follow math from Pythagoras to Einstein and beyond. It all leads to the ultimate riddle: Is math a human invention or the discovery of the language of the universe?

The Great Math Mystery. PBS NOVA

Math is invisible. Similar to the tree falling in the forest, there are people who believe that if no person existed to count, math wouldn’t be around. But is this true? Or if math is a real entity that exists, are there formulas and mathematical concepts out there in the universe that are undiscovered?

Is Math a Feature of the Universe or a Feature of Human Creation? PBS 8 minute video

How is it possible that everything in the physical world is describable by math? Eugene Wigner writes “…The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.”

Or as Albert Einstein said, “The most incomprehensible thing about the universe is that it is comprehensible.” Let’s explore this subject further in this essay:

The Unreasonable Effectiveness of Mathematics in the Natural Sciences, by Eugene Wigner

Mathematics is the language of physics. Physical principles and laws, which would take two or even three pages to write in words, can be expressed in a single line using mathematical equations. Such equations, in turn, make physical laws more transparent, interpretation of physical laws easier, and further predictions based on the laws straightforward. – Mesfin Woldeyohannes,

## Our articles

Math is the language of physics

Math skills used in high school physics

Applied math: The future of photography on phones depends on coding

Basic mathematics

What does it mean to divide a fraction by a fraction?

Geometry

All orbits are conic sections.

Calculus

Vector Calculus: Understanding the Dot Product

Probability and Statistics

The Black Swan, by Nassim Taleb

Number theory

In what sense are “imaginary” numbers real, or useful?

## Math in art and poetry

Mathematics and science are not alternatives to the art, poetry or literature – they may in fact be subjects of inspiration for artists – or indeed be tools by which the artist creates new works of art. Colleges may offer classes such as “Mathematics in Art and Architecture”

The medieval Spanish-Jewish poet and philosopher, Solomon ibn Gabriol, uses mathematics as a way to describe God’s one-ness in his Keter Malchut, The Royal Crown. (1tth century) Translation by Raphael Loewe.

Thy Name is One – of all the primes the Prime,

Base of all algebraic argument,

A Unity beyond account, sublime,

That leaves the schoolmen lost in wonderment:

Uniquity, that neither wanes nor grows,

No plus, no defect knows:

Oneness not gained from accident, nor told,

On which no change, no factor may impose

Nor attribute, nor surrogate; to hold

In logic’s bounds that Oneness strict defined

Eludes my wearied mind….

Physicist Werner Heisenberg said, “When I meet God, I am going to ask him two questions: why relativity? And why turbulence? I really believe he will have an answer for the first.” As difficult as turbulence is to understand mathematically, we can use art to depict the way it looks. Natalya St. Clair illustrates how Van Gogh captured this deep mystery of movement, fluid and light in his work: The unexpected beauty and math behind Van Gogh’s “Starry Night”

## Engineering: Using math to build

### Our introduction to the topic of engineering

You may be interested in reading Mathematics and art (Wikipedia) , Is mathematics beautiful?, and Why the history of maths is also the history of art

Inga Nielsen / Alex Landa / Shutterstock

## External links

Philosophy of Mathematics (Wikipedia)

Is the Universe Made of Math? Max Tegmark

Why is geometry mathematics and not physics?

## Learning Standards

Mathematics: Massachusetts Curriculum Frameworks

Common Core Mathematics Standards

## Post Credits Scene 😉

Gauss

Euler

Newton

## Quotes

SILENCE following this discourse, Diogenianus began and said: Since our discourse is about the Gods, shall we, especially on his own birthday, admit Plato to the conference, and enquire upon what account he says (supposing it to be his sentence) that **God always plays the geometer**? I said that this sentence was not plainly set down in any of his books; yet there are good arguments that it is his, and it is very much like his expression. Tyndares presently subjoining said: Perhaps, Diogenianus, you imagine that this sentence intimates some curious and difficult speculation, and not that which he hath so often mentioned, when he praiseth geometry as a science that takes off men from sensible objects, and makes them apply themselves to the intelligible and eternal Nature, the contemplation of which is the end of philosophy, as a view of the mysteries of initiation into holy rites. For the nail of pain and pleasure, that fastens the soul to the body, seems to do us the greatest mischief, by making sensible things more powerful over us than intelligible, and by forcing the understanding [p. 403] to determine rather according to passion than reason. For the understanding, being accustomed by the vehemency of pain or pleasure to be intent on the mutable and uncertain body, as if it really and truly were, grows blind as to that which really is, and loses that instrument and light of the soul, which is worth a thousand bodies, and by which alone the Deity can be discovered. Now in all sciences, as in plain and smooth mirrors, some marks and images of the truth of intelligible objects appear, but in geometry chiefly; which, according to Philo, is the chief and principal of all, and doth bring back and turn the understanding, as it were, purged and gently loosened from sense. And therefore Plato himself dislikes Eudoxus, Archytas, and Menaechmus for endeavoring to bring down the doubling the cube to mechanical operations; for by this means all that was good in geometry would be lost and corrupted, it falling back again to sensible things, and not rising upward and considering immaterial and immortal images, in which God being versed is always God.

– **Plutarch.** Plutarch’s Morals. Translated from the Greek by several hands. Corrected and revised by William W. Goodwin, PH. D. Boston. Little, Brown, and Company. Cambridge. Press Of John Wilson and son. 1874. 3.