Do the numbers.

I’ve always loved mathematics to approximately the same extent that I consider myself a math rookie when it comes to it’s higher forms — which is a lot.  With regard to my lack of expertise in math I take some level of comfort from the words of Albert Einstein when he said: “Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.”  There is a true beauty in Math in the way it works, the way it holds together. Perhaps the most wondrous aspect of math for me is the way that it can be used to describe the physical reality that we observe all around us and the way that following the leads set out by mathematics result in fantastic discoveries.  These two features of mathematics are highlighted in the following article about the relationship of math with astronomy which I believe you may find interesting.  This blog post by Joshua Carroll appeared at www.universetoday.com as: Mathematics: The Beautiful Language of the Universe.  RMF

Mathematics: The Beautiful Language of the Universe
by JOSHUA CARROLL

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Joshua Carroll

Joshua Carroll

Let us discuss the very nature of the cosmos. What you may find in this discussion is not what you expect. Going into a conversation about the universe as a whole, you would imagine a story full of wondrous events such as stellar collapse, galactic collisions, strange occurrences with particles, and even cataclysmic eruptions of energy. You may be expecting a story stretching the breadth of time as we understand it, starting from the Big Bang and landing you here, your eyes soaking in the photons being emitted from your screen. Of course, the story is grand. But there is an additional side to this amazing assortment of events that oftentimes is overlooked; that is until you truly attempt to understand what is going on. Behind all of those fantastic realizations, there is a mechanism at work that allows for us to discover all that you enjoy learning about. That mechanism is mathematics, and without it the universe would still be shrouded in darkness. In this article, I will attempt to persuade you that math isn’t some arbitrary and sometimes pointless mental task that society makes it out to be, and instead show you that it is a language we use to communicate with the stars.  

Screen Shot 2015-07-12 at 7.01.23 PM  We are currently bound to our solar system. This statement is actually better than it sounds, as being bound to our solar system is one major step up from being bound simply to our planet, as we were before some very important minds elected to turn their geniuses toward the heavens. Before those like Galileo, who aimed his spyglass towards the sky, or Kepler discovering that planets move about the sun in ellipses, or Newton discovering a gravitational constant, mathematics was somewhat limited, and our understanding of the universe rather ignorant. At its core, mathematics allows a species bound to its solar system to probe the depths of the cosmos from behind a desk. Now, in order to appreciate the wonder that is mathematics, we must first step back and briefly look at its beginnings and how it is integrally tied into our very existence.

Screen Shot 2015-07-12 at 7.03.42 PM  Mathematics almost certainly came about from very early human tribes (predating Babylonian culture which is attributed to some of the first organized mathematics in recorded history), that may have used math as a way of keeping track of lunar or solar cycles, and keeping count of animals, food and/or people by leaders. It is as natural as when you are a young child and you can see that you have one toy plus one other toy, meaning you have more than one toy. As you get older, you develop the ability to see that 1+1=2, and thus simple arithmetic seems to be interwoven into our very nature. Those that profess that they don’t have a mind for math are sadly mistaken because just as we all have a mind for breathing, or blinking, we all have this innate ability to understand arithmetic. Mathematics is both a natural occurrence and a human designed system. It would appear that nature grants us this ability to recognize patterns in the form of arithmetic, and then we systematically construct more complex mathematical systems that aren’t obvious in nature but let us further communicate with nature.

All this aside, mathematics developed alongside of human development, and carried on similarly with each culture that was developing it simultaneously. It’s a wonderful observation to see that cultures that had no contact with one another were developing similar mathematical constructs without conversing. However, it wasn’t until mankind decidedly turned their mathematical wonder towards the sky that math truly began to develop in an astonishing way. It is by no mere coincidence that our scientific revolution was spurred by the development of more advanced mathematics built not to tally sheep or people, but rather to further our understandings of our place within the universe. Once Galileo began measuring the rates at which objects fell in an attempt to show mathematically that the mass of an object had little to do with the speed in which it fell, mankind’s future would forever be altered.

Screen Shot 2015-07-12 at 7.05.40 PM  This is where the cosmic perspective ties in to our want to further our mathematical knowledge. If it were not for math, we would still think we were on one of a few planets orbiting a star amidst the backdrop of seemingly motionless lights. This is a rather bleak outlook today compared to what we now know about the awesomely large universe we reside in. This idea of the universe motivating us to understand more about mathematics can be inscribed in how Johannes Kepler used what he observed the planets doing, and then applied mathematics to it to develop a fairly accurate model (and method for predicting planetary motion) of the solar system. This is one of many demonstrations that illustrate the importance of mathematics within our history, especially within astronomy and physics.

Screen Shot 2015-07-12 at 7.07.17 PM  The story of mathematics becomes even more amazing as we push forward to one of the most advanced thinkers humanity has ever known. Sir Isaac Newton, when pondering the motions of Halley’s Comet, came to the realization that the math that had been used thus   far to describe physical motion of massive bodies, simply would not suffice if we were to ever understand anything beyond that of our seemingly limited celestial nook. In a show of pure brilliance that lends validity to my earlier statement about how we can take what we naturally have and then construct a more complex system upon it, Newton developed the Calculus in which this way of approaching moving bodies, he was able to accurately model the motion of not only Halley’s comet, but also any other heavenly body that moved across the sky.

In one instant, our entire universe opened up before us, unlocking almost unlimited abilities for us to converse with the cosmos as never before. Newton also expanded upon what Kepler started. Newton recognized that Kepler’s mathematical equation for planetary motion, Kepler’s 3rd Law ( P2=A3 ), was purely based on empirical observation, and was only meant to measure what we observed within our solar system. Newton’s mathematical brilliance was in realizing that this basic equation could be made universal by applying a gravitational constant to the equation, in which gave birth to perhaps one of the most important equations to ever be derived by mankind; Newton’s Version of Kepler’s Third Law.

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You can still see where Kepler’s 3rd Law remains, but with the added values of the gravitational constant G, and M and m representing the masses of the two bodies in question, this equation is no longer restricted to just our solar system
You can still see where Kepler’s 3rd Law remains, but with the added values of the gravitational constant G, and M and m representing the masses of the two bodies in question, this equation is no longer restricted to just our solar system
What Newton realized was that when things move in non-linear ways, using basic Algebra would not produce the correct answer. Herein lays one of the main differences between Algebra and Calculus. Algebra allows one to find the slope (rate of change) of straight lines (constant rate of change), whereas Calculus allows one to find the slope of curved lines (variable rate of change). There are obviously many more applications of Calculus than just this, but I am merely illustrating a fundamental difference between the two in order to show you just how revolutionary this new concept was. All at once, the motions of planets and other objects that orbit the sun became more accurately measurable, and thus we gained the ability to understand the universe a little deeper. Referring back to Netwon’s Version of Kepler’s Third Law, we were now able to apply (and still do) this incredible physics equation to almost anything that is orbiting something else. From this equation, we can determine the mass of either of the objects, the distance apart they are from each other, the force of gravity that is exerted between the two, and other physical qualities built from these simple calculations.

With his understanding of mathematics, Newton was able to derive the aforementioned gravitational constant for all objects in the universe ( G = 6.672×10-11 N m2 kg-2 ). This constant allowed him to unify astronomy and physics which then permitted predictions about how things moved in the universe. We could now measure the masses of planets (and the sun) more accurately, simply according to Newtonian physics (aptly named to honor just how important Newton was within physics and mathematics). We could now apply this newfound language to the cosmos, and begin coercing it to divulge its secrets. This was a defining moment for humanity, in that all of those things that prohibited our understandings prior to this new form of math were now at our fingertips, ready to be discovered. This is the brilliance of understanding Calculus, in that you are speaking the language of the stars.

There perhaps is no better illustration of the power that mathematics awarded us then in the discovery of the planet Neptune. Up until its discovery in September of 1846, planets were discovered simply by observing certain “stars” that were moving against the backdrop of all the other stars in odd ways. The term planet is Greek for “wanderer”, in that these peculiar stars wandered across the sky in noticeable patterns at different times of the year. Once the telescope was first turned upwards towards the sky by Galileo, these wanderers resolved into other worlds that appeared to be like ours. If fact, some of these worlds appeared to be little solar systems themselves, as Galileo discovered when he began recording the moons of Jupiter as they orbited around it.

After Newton presented his physics equations to the world, mathematicians were ready and excited to begin applying them to what we had been keeping track of for years. It was as if we were thirsty for the knowledge, and finally someone turned on the faucet. We began measuring the motions of the planets and gaining more accurate models for how they behaved. We used these equations to approximate the mass of the Sun. We were able to make remarkable predictions that were validated time and again simply by observation. What we were doing was unprecedented, as we were using mathematics to make almost impossible to know predictions that you would think we could never make without actually going to these planets, and then using actual observation to prove the math correct. However, what we also did was begin to figure out some odd discrepancies with certain things. Uranus, for instance, was behaving not as it should according to Newton’s laws.

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Here you can see that the inner planet is being perturbed by the outer planet, in our situation, that outer planet was Neptune, not yet discovered.
Here you can see that the inner planet is being perturbed by the outer planet. In our situation, that outer planet was Neptune, which had yet to be discovered.
What makes the discovery of Neptune so wonderful was the manner in which it was discovered. What Newton had done was uncover a deeper language of the cosmos, in which the universe was able to reveal more to us. And this is exactly what happened when we applied this language to the orbit of Uranus. The manner in which Uranus orbited was curious and did not fit what it should have if it was the only planet that far out from the sun. Looking at the numbers, there had to be something else out there perturbing its orbit. Now, before Newton’s mathematical insights and laws, we would have had no reason to suspect anything was wrong in what we observed. Uranus orbited in the way Uranus orbited; it was just how it was. But, again revisiting that notion of mathematics being an ever increasing dialogue with the universe, once we asked the question in the right format, we realized that there really must be something else beyond what we couldn’t see. This is the beauty of mathematics writ large; an ongoing conversation with the universe in which more than we may expect is revealed.

Screen Shot 2015-07-12 at 7.11.57 PM  It came to a French mathematician Urbain Le Verrier who sat down and painstakingly worked through the mathematical equations of the orbit of Uranus. What he was doing was using Newton’s mathematical equations backwards, realizing that there must be an object out there beyond the orbit of Uranus that was also orbiting the sun, and then looking to apply the right mass and distance that this unseen object required for perturbing the orbit of Uranus in the way we were observing it was. This was phenomenal, as we were using parchment and ink to find a planet that nobody had ever actually observed. What he found was that an object, soon to be Neptune, had to be orbiting at a specific distance from the sun, with the specific mass that would cause the irregularities in the orbital path of Uranus. Confident of his mathematical calculations, he took his numbers to the New Berlin Observatory, where the astronomer Johann Gottfried Galle looked exactly where Verrier’s calculations told him to look, and there lay the 8th and final planet of our solar system, less than 1 degree off from where Verrier’s calculations said for him to look. What had just happened was an incredible confirmation of Newton’s gravitational theory and proved that his mathematics were correct.

Screen Shot 2015-07-12 at 7.14.04 PM Neptune is more than just the 8th planet in our solar system; it is a celestial reminder of the power that mathematics grants us.
These types of mathematical insights continued on long after Newton. Eventually, we began to learn much more about the universe with the advent of better technology (brought about by advances in mathematics). As we moved into the 20th century, quantum theory began to take shape, and we soon realized that Newtonian physics and mathematics seemed to hold no sway over what we observed on the quantum level. In another momentous event in human history, yet again brought forth by the advancement in mathematics, Albert Einstein unveiled his theories of General and Special Relativity, which was a new way to look not only at gravity, but also on energy and the universe in general. What Einstein’s mathematics did was allow for us to yet again uncover an even deeper dialogue with the universe, in which we began to understand its origins.

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Continuing this trend of advancing our understandings, what we have realized is that now there are two sects of physics that do not entirely align. Newtonian or “classical” physics, that works extraordinarily well with the very large (motions of planets, galaxies, etc…) and quantum physics that explains the extremely small (the interactions of sub-atomic particles, light, etc…). Currently, these two areas of physics are not in alignment, much like two different dialects of a language. They are similar and they both work, but they are not easily reconcilable with one another. One of the greatest challenges we face today is attempting to create a mathematical grand “theory of everything” which either unites the laws in the quantum world with that of the macroscopic world, or to work to explain everything solely in terms of quantum mechanics. This is no easy task, but we are striving forward nonetheless.

As you can see, mathematics is more than just a set of vague equations and complex rules that you are required to memorize. Mathematics is the language of the universe, and in learning this language, you are opening yourself up the core mechanisms by which the cosmos operates. It is the same as traveling to a new land, and slowly picking up on the native language so that you may begin to learn from them. This mathematical endeavor is what allows us, a species bound to our solar system, to explore the depths of the universe. As of now, there simply is no way for us to travel to the center of our galaxy and observe the supermassive black hole there to visually confirm its existence. There is no way for us to venture out into a Dark Nebula and watch in real time a star being born. Yet, through mathematics, we are able to understand how these things exist and work. When you set about to learn math, you are not only expanding your mind, but you are connecting with the universe on a fundamental level. You can, from your desk, explore the awesome physics at the event horizon of a black hole, or bear witness to the destructive fury behind a supernova. All of those things that I mentioned at the beginning of this article come into focus through mathematics. The grand story of the universe is written in mathematics, and our ability to translate those numbers into the events that we all love to learn about is nothing short of amazing. So remember, when you are presented with the opportunity to learn math, accept every bit of it because math connects us to the stars.

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About Joshua Carroll
I am currently an aspiring student of science. I will be working towards my undergraduate degree in Earth and Space Physics with a Minor in Astronomy. My future goals are to complete a Masters and then a Ph.D in Astrophysics. As well as studying the astro-sciences, I am also a three-time combat vet. Having completed my service, I now seek to advance the public’s knowledge on astronomical phenomena.

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Comments on this entry are closed.

btraymd June 6, 2015, 10:47 PM
I certainly agree that mathematics can be used to accurately describe many observations. However, there has been a colossal misuse of math today. It has been used to create reality instead of describe it. The best examples are the mathematical creation of dark matter and black holes.
Neither of these concepts has ever been observed and they defy confirmation. They are merely mathematical constructs. What brought about the need for these constructs? The fact that astrophysicists and cosmologists have ruled that gravity must be the driving force for galaxy formation. When it was determined that gravity had insufficent forces to cause galaxy formation and maintain galactic structure the need for other forces (unseen and unknown, of course) was apparent. And oila, we have dark matter and black holes. Even with these add-ons intended to salvage gravity-based theories there are still inadequate forces.
The use of math to salvage failed theories has led the field of cosmology down a dead end road.
Only now with the newest technology and advances in radio telescopes do we see the real forces behind star and galaxy formation. Massive magnetic fields are now detected surrounding “black holes” which are now more correctly being referred to as galactic centers. It is becoming more likely that these centers are super dense plasmoids. There are galactic sized magnetic fields surrounding galaxies as well. Intergalactic electric currents (“rivers of hydrogen”) connecting stars and galaxies have been identified and it is likely that these currents are associated with equally vast magnetic fields.
At Los Alamos National Lab, Anthony Pratt used plasma, magnetic fields and electric current to demonstrate spiral galaxy formation in his plasma physics lab. AND these lab induced “spiral galaxies” had the necessary velocities to maintain their shape. Black holes and dark matter were not required. These observations could be and were mathematically described.
It seems that we may be leaving the age of Newton and entering the age of Faraday and Tesla. Progress can now be made in cosmology towards rational explanations of observations without the need for imaginary mathematical constructs.
The use of the failed gravity based theories to explain the newest observations has resulted in “mysteries” and contradictions. When observed with consideration given to electromagnetic principles the mysteries disappear.
I think the presentation by Donald Scott to NASA at the Goddard Colloquia on Engineering in 2009 is a good starting point for one to review the contributions that can be made to the field of cosmology by electrical engineers and plasma physicists. There are many that he references in this presentation that have done ground breaking work in the fields of electromagnetism and plasma physics that can be directly applied to the latest observations from radio telescopes.
This work can clearly be used to explain “confounding mysteries” that exist in the field of cosmology. Without knowledge of a “Z pinch”, astrophysicists are left guessing about routine electrical events.
Math can then be used to confirm these observations and electromagnetic based explanations. Science fiction can then be left to the novelists.

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DrFlimmer June 7, 2015, 4:51 AM
No. It’s not.

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Olaf June 7, 2015, 7:21 AM
Nope, the electric universe still does not exist.

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FarAwayLongAgo June 7, 2015, 7:23 AM
I recommend public lectures at Canadian Institute for Theoretical Astrophysics which are available on youtube here: http://www.youtube.com/user/citaseminars They have more than a dozen talks about astronomical magnetic fields. I think it has been a neglected topic because it has been hard to observe. But nowadays it is possible to map large scale magnetic field lines by observing the polarization of photons (something like that). But I never heard anyone of them talk about how that could somehow replace gravity or dark matter or affecting the rotation of galaxies. There is indeed an “electric universe” of sorts, but it does not conflict with established physics. Dark matter is carefully mapped. Black holes are observed.

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Jeffrey Boerst June 7, 2015, 10:16 AM
Me and… me and my very, very small but tight-knit internet community knows more… more about outer space stuff than all the outer space science guys out there and… and… and I’m just a-gonna keep… keep a-coming back here and letting you all know over and over and over and over again because it shows the whole wide WORLD how much more smarter I am than… than all of you guys put toGETHer! ….Ya…! You’ll see…. You’ll ALL see someday. You WATCH!

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mewo June 8, 2015, 2:20 AM
Typical misconception. Dark matter was not just conjured up by some mathematician one day as a whim. It was postulated as a way of explaining observations, and only after all other hypotheses failed for one reason after another. That includes all weird appeals to magnetic or electric fields.

Scientists do not use mathematics to “create reality”, but to describe what they see.

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Warlord0616 June 7, 2015, 3:31 AM
I appreciate the awesome article. Thank you for your military also.

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Chan PN June 7, 2015, 8:59 AM
I second that.

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joseluis7696 June 7, 2015, 5:18 AM
Very clear and informing article. Congratulations to its author. But I cannot agree with readers comments based on the “Electric Universe” theory. That theory has no serious base and is not supported by correct math, not even by a thorough understanding of electromagnetic theory.
Not to mention that the existence of singularities has nothing to do with the gravitational stability of galaxies.

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FarAwayLongAgo June 7, 2015, 6:56 AM
Oh, they don’t care about being wrong. They just feel that they need something for themselves to believe in.

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Olaf June 7, 2015, 7:30 AM
“Mathematics is the language of the universe”

This is actually not true.

The universe has a completely different language that mathematics. But mathematics is generic enough to be uses to describe and predict simplified models (e.g. 4 forces) of this universe. Mathematics can also be used to describe pure fictional worlds like used in gaming.

It is up to experiments to determine what mathematical formulas really exists in this universe and which are fictional.

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JustinSky June 7, 2015, 10:08 AM
Well said!

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FarAwayLongAgo June 7, 2015, 2:53 PM
But math is just logic. Physical reality is logical. Math looks generic because math in and of itself uses numbers and anonymous variables like “x”, precisely for the purpose of allowing maximum degrees of freedom. But once you use measured physical entities and use them mathematically, then math reveals what is true about the physical reality beyond what was measured. Even if you logically use physically impossible symbols, like negative numbers, during the calculation.

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BlackWolfStanding June 7, 2015, 9:37 AM
Great Article! And no buts about it.

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JustinSky June 7, 2015, 10:06 AM
This is an excellent article!

Galileo wrote, “La mathematca è l’ alfabeto nel quale DIO ha scritto l’ universo.” “Mathematics is the alphabet in which God wrote the universe. The alphabet, not the language.

Some scientists marvel at how mathematical the universe seems to be. But the nature of the universe does not necessarily flow from the mathematics. Mathematics is very flexible and the mathematics is chosen to fit the observations. Once a good fit is found it is then tested to see if it makes valid predictions. We must not assume the underlying nature of the universe is mathematical. Mathematics is just very powerful in representing it. The randomness of some quantum mechanical events may be the point where mathematics fails to mirror the real nature of the universe.

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BlackWolfStanding June 7, 2015, 11:12 AM
Maybe math might not be the problem but the user?

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UFOsMOTHER June 8, 2015, 10:02 AM
Reality is Reality twist it any way you want if it makes you happy but that does Not make it Real 🙂 Amen

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Ynot594123 June 8, 2015, 5:06 PM
Nice article, but once again as previously stated, without physically observing something in its true mult dimension universe we live in,the mathematical principles used to describe our cosmos are at the reins of the observer, thus allowing personal beliefs and persuasion. Truth be told that just because one mathematical model fits one answer doesnt mean that theres only one answer but rather there can still be multi answers of movement for that particular mathematical model. For example how and why do we teach and continue to believe facts that mathematical models used to describe movements in our galaxy and solar system in a 2 dimensional platform when we know we live in a multi dimension universe and galaxy. Just cause those mathematical principles fit those porpotions doesnt exclude milti answers in fact its much easier to fit a complex mathematical principle in a multi dimension into a simple 2 dimensional model than it is to do the opposite, which brings us to the reality that jyst like history, change is a constant and to solely rely on one apparatus like math to hold as absolute truth is just as ignorant as believing that the world is flat just cause its the simplest answer. We need to stop believing that theres only one way. Our current knowledge of the cosmos is still very immature and new discoveries everday are changing past theories that mathematical principles used worked but now have to be changed. As long as people want be self rightous and ego centric and so fixated on past theories thought to be true and excepted and take offense to new ways, then we are just
Like the Roman Church that drove us into the dark ages and blocked our current knowledge from existed. Please people who consist of our scientific astronomy fans, stop repeating the mistakes of the past and stop being so fixated on your own views as right and CHANGE your mind to be accepting to change because thats the only way truths will be unveiled and we can progressive forward. I think its sad that a website named universetoday can be so self rightous on their views and be willingly to accept only mathematical principles and models as absolute truths kniwing that history as always proven this wrong.

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About ronfurg

Former naval officer, federal investigator, forensic scientist, senior executive service member and pastor. In retirement serves as volunteer and life group leader at New Life Christian Church (www.newlife4me.com). Devoted to beautiful wife, kids and grandkids. Looking forward to the time when every knee will bow and every tongue will confess that Jesus Christ is Lord of all and that He is the Way, the Truth, and the Life.
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