Towards the end of the 17th century, Newton and Leibniz formulated calculus, the branch of mathematics dealing with change. Calculus is the Latin word for a small stone, like those used to aid calculations. It is ironic that small stones, or calculi, were the downfall of both Newton and Leibniz. Calculus gave us a means of understanding the motion of heavenly bodies and of mechanical systems on Earth.
Using these equations, Maxwell predicted the existence of radio waves, which were generated and detected experimentally by Heinrich Hertz about 20 years later. Attitudes of professional mathematicians to the nature of mathematics cover a wide spectrum. Formalists view it as a mechanistic process, deducing theorems from initial assumptions, always following clear logical steps.
To the formalist, mathematics is not an abstract representation of reality, but is more like a game with clearly defined rules but no deep underlying meaning. In contrast, the Platonic view holds that mathematical concepts are eternal and unchanging.
This in turn was preceded by the Pythagorean view that reality is essentially numerical in nature. A recent book, Our Mathematical Universe, by Swedish cosmologist Max Tegmark , maintains the universe is not just well described by mathematics; it is mathematics. We still have much to learn about the ultimate nature of reality. He blogs at thatsmaths. It is denied as a serious problem but could cause reputational damage.
The starting point for reverse maths is a base theory that is strong enough to state the theorems of interest, but not strong enough to prove them. Further global warming will cause more trouble, but there is much we can still do. Italian example shows how investment in rail can drastically cut traffic on air routes.
Science fiction writers give us a glimpse into the possible worlds that await us. Stuart Mathieson: International efforts can have an effect within our lifetimes.
Here again, human-made designs like the rectangular shape of this book don't count. But try throwing a pebble and watch the beautiful shape that nature makes for its trajectory! The trajectories of anything you throw have the same shape, called an upside-down parabola. When we observe how things move around in orbits in space, we discover another recurring shape: the ellipse.
Moreover, these two shapes are related: the tip of a very elongated ellipse is shaped almost exactly like a parabola, so in fact, all of these trajectories are simply parts of ellipses. We humans have gradually discovered many additional recurring shapes and patterns in nature, involving not only motion and gravity, but also areas as disparate as electricity, magnetism, light, heat, chemistry, radioactivity, and subatomic particles.
These patterns are summarized by what we call our laws of physics. Just as the shape of an ellipse, all these laws can be described using mathematical equations. Equations aren't the only hints of mathematics that are built into nature: there are also numbers. As opposed to human creations like the page numbers in this book, I'm now talking about numbers that are basic properties of our physical reality. For example, how many pencils can you arrange so that they're all perpendicular at 90 degrees to each other?
Where did that number 3 come sailing in from? We call this number the dimensionality of our space, but why are there 3 dimensions rather than 4 or 2 or 42? And why are there, as far as we can tell, exactly 6 kinds of quarks in our Universe? There are also numbers encoded in nature that require decimals to write out — for example, the proton about From just 32 such numbers, we physicists can in principle compute every other physical constant ever measured.
There's something very mathematical about our Universe, and that the more carefully we look, the more math we seem to find. So what do we make of all these hints of mathematics in our physical world? Most of my physics colleagues take them to mean that nature is for some reason described by mathematics, at least approximately, and leave it at that. But I'm convinced that there's more to it, and let's see if it makes more sense to you than to that professor who said it would ruin my career.
The mathematical universe hypothesis I was quite fascinated by all these mathematical clues back in grad school. One Berkeley evening in , while my friend Bill Poirier and I were sitting around speculating about the ultimate nature of reality, I suddenly had an idea for what it all meant: that our reality isn't just described by mathematics — it is mathematics, in a very specific sense.
Not just aspects of it, but all of it, including you. My starting assumption, the external reality hypothesis, states that there exists an external physical reality completely independent of us humans. It's important to remember, however, that it's we humans who create these concepts; in principle, everything could be calculated without this baggage. For that, some tweaks needed to be made to the math, creating what some call a rigged Hilbert space or a physical Hilbert space. The mathematical rules needed to be applied with certain specific caveats, or the results of our physical Universe would never be recoverable.
The neutral Higgs field gray square breaks the electroweak symmetry and interacts with other particles to give them mass. This diagram shows the structure of particles, but is rooted in both mathematics and physics. Today, it's grown very fashionable in theoretical physics to appeal to mathematics as a potential way forward to an even more fundamental theory of reality.
A number of mathematical-based approaches have been tried over the years:. These mathematical explorations are interesting and potentially relevant for physics: they may hold clues as to what secrets the Universe might have in store beyond what's presently known. But mathematics alone cannot teach us how the Universe works.
We will obtain no definitive answers without confronting its predictions with the physical Universe itself. Visualizing the multiplication of the unit octonions, of which there are 8, requires thinking in The multiplication table for any two unit octonions is also shown right.
Octonions are a fascinating mathematical structure, but offers non-unique solutions to a myriad of possible applications. In some ways, it's a lesson that every physics student learns the first time they calculate the trajectory of an object thrown into the air. How far does it go? Where does it land? How long does it spend in the air? When you solve the mathematical equations — Newton's equations of motion — that govern these objects, you don't get "the answer.
But in reality, there's only one object. It only follows one trajectory, landing in one location at one specific time. Which answer corresponds to reality? Mathematics won't tell you. For that, you need to understand the particulars of the physics problem in question, as only that will tell you which answer has a physical meaning behind it. Mathematics will get you very far in this world, but it won't get you everything. Without a confrontation with reality, you cannot hope to understand the physical Universe.
This is a BETA experience. You may opt-out by clicking here. All matter is made up of particles, which have properties such as charge and spin, but these properties are purely mathematical, he says. And space itself has properties such as dimensions, but is still ultimately a mathematical structure. But if the universe really is mathematics, he added, "There's nothing we can't, in principle, understand. The idea follows the observation that nature is full of patterns, such as the Fibonacci sequence , a series of numbers in which each number is the sum of the previous two numbers.
The flowering of an artichoke follows this sequence, for example, with the distance between each petal and the next matching the ratio of the numbers in the sequence. The nonliving worldalso behaves in a mathematical way.
If you throw a baseball in the air, it follows a roughly parabolic trajectory.
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