# The evolution of mathematics, from agriculture to quantum mechanics

Why is math the universal language? NASA's Michelle Thaller solves that one.

Dr. Michelle Thaller is an astronomer who studies binary stars and the life cycles of stars. She is Assistant Director of Science Communication at NASA. She went to college at Harvard University, completed a post-doctoral research fellowship at the California Institute of Technology (Caltech) in Pasadena, Calif. then started working for the Jet Propulsion Laboratory's (JPL) Spitzer Space Telescope. After a hugely successful mission, she moved on to NASA's Goddard Space Flight Center (GSFC), in the Washington D.C. area. In her off-hours often puts on about 30lbs of Elizabethan garb and performs intricate Renaissance dances. For more information, visit NASA.

**MICHELLE THALLER:** Oğuzhan, you asked, why is mathematics the universal language? And this is something I've actually thought a lot about. Mathematics is in some ways kind of scary in how useful it is at really describing how the universe works around us. Now, to give you an idea, the origin of mathematics seems very straightforward. We can count on our fingers up to 10, and maybe it was useful to understand how many sheep you had? So you could start counting sheep and then you either added or subtracted sheep as you got more or as you lost some. It was a simple thing. We learned how to count. We learned how to add and subtract. The idea of multiplying and dividing is a little more abstract, but that also makes sense. That's something that we can kind of visualize.

But then what amazes me is that this led us on a tremendously complicated journey that's still going on to this day. And we had no idea where this would lead us. If you can do multiplication and subtraction, it's not too long before you begin to develop the basic building blocks of calculus. And calculus describes how moving objects can change, how things can accelerate. If you want to describe an apple falling from a tree to the ground or a ball rolling down a hill, that's calculus. It's the mathematics of how things can change over time. That's really interesting, and the amazing thing is it works so well. If you use these equations to predict how a ball will roll down a hill, reality matches that. It really does tell you how something is going to behave. So now we've gone from counting on our fingers how many sheep we have to being able to predict what the universe around us is going to do. That's incredibly powerful.

Now we look around us and we see things like planets orbiting the stars or the galaxy turning around, and we realize those equations of motion apply to everything else in the universe. It's not just here. It's not just on the surface of the Earth, but we can look at things literally billions of light years out in space, and they're following those same rules of mathematics. But now things got strange. We started to play with calculus. We started to see where it would go. What happens if you put in more variables and you solve for more things at once? And we end up with some very strange abstract concepts that turned out to be surprisingly useful. One of the things that kind of worries me is something called imaginary numbers. Imaginary numbers are numbers that don't really make sense in our proper definition of mathematics. Take, for example, the square root of negative 1. Now, in mathematics, if you multiply something by itself it always turns out to be a positive number. That's never a negative number. But somebody said, what happens if we start to do the mathematics of how an imaginary number -- this can't be real. The square root of negative 1 doesn't make any sense. But it turns out to be able to describe how things rotate, and that became the foundation of quantum mechanics. And here's the thing, now when you use a number that shouldn't exist -- that doesn't make any sense -- it predicts exactly how an atom will vibrate, It will predict how quantum mechanics at a very small scale runs, and it needs a type of math that doesn't make any real sense to us but it works. It works perfectly.

So we keep getting led farther and farther down this rabbit hole. Where does math lead us? Now we realize that you can describe physics incredibly well if you allow the universe to exist in many different dimensions-- more than three dimensions that we're familiar with. In fact, specifically, if you want to do particle physics, it requires 11 dimensions. That's not something our minds comprehend, but we can do the math. We can do the math of how things would behave if they could move in 11 different directions. And it turns out to predict exactly the results we get from particle physics. That's kind of scary. Does that mean that's real? Are there really 11 dimensions? The math works so well, and the predictions are so strong that it can't just be nonsense. But now we've gone to the limit of what I can tell you; is it real or not? Our math has given us something incredibly useful, but it's taken us completely out of our realm of common sense, of human scale of how our minds work and even our sense of space and time. I don't think that journey's over yet. Where is math going to lead us? It may lead us to understand things like the universe is a type of a hologram? That was a mathematical solution to How things work around a black hole, and it works really, really well. So I think it's wonderful and a little bit scary that you start counting on your fingers. You get to 11 dimensions of space and time. And where else?

- Mathematics has snowballed from counting to 10 on our fingers, to calculus, to abstract concepts like imaginary numbers that move in 11 dimensions and predict particles physics.
- The math that led us down the rabbit hole of quantum mechanics is bizarre and while we can crunch the numbers, we can't really understand what they mean.
- If the math confirms that particles can move in 11 dimensions, is that a fundamental truth of the universe?

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## New Hubble images add to the dark matter puzzle

The images and our best computer models don't agree.

### A trio of intriguing galaxy clusters

<img type="lazy-image" data-runner-src="https://assets.rebelmouse.io/eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpbWFnZSI6Imh0dHBzOi8vYXNzZXRzLnJibC5tcy8yNDQzNDA0OS9vcmlnaW4uanBnIiwiZXhwaXJlc19hdCI6MTYxNTkzNzUyOH0.0IRzkzvKsmPEHV-v1dqM1JIPhgE2W-UHx0COuB0qQnA/img.jpg?width=980" id="d69be" class="rm-shortcode" data-rm-shortcode-id="2d2664d9174369e0a06540cb3a3a9079" data-rm-shortcode-name="rebelmouse-image" />The three galaxy clusters imaged for the study

<p>The discrepancy has to do with images of three galaxy clusters captured by Hubble's <a href="https://www.spacetelescope.org/about/general/instruments/wfc3/" target="_blank">Wide Field Camera 3</a> and <a href="https://www.spacetelescope.org/about/general/instruments/acs/" target="_blank">Advanced Camera for Surveys</a> as part of two Hubble projects: The Frontier Fields and the Cluster Lensing And Supernova survey with Hubble (CLASH) programs. The three clusters are called <a href="https://spacetelescope.org/images/heic1115a/" target="_blank">MACS J1206.2-0847</a>, <a href="https://www.spacetelescope.org/images/heic1416a/" target="_blank">MACS J0416.1-2403</a>, and <a href="https://www.spacetelescope.org/images/heic1615a/" target="_blank">Abell S1063</a>.</p><p>Such imagery can be used for authenticating — or exposing flaws in —predictive computer models of dark matter's behavior, locations, and concentrations. </p><p>Lead author <a href="https://www.unibo.it/sitoweb/massimo.meneghetti/en" target="_blank" rel="noopener noreferrer">Massimo Meneghetti</a> of the INAF-Observatory of Astrophysics and Space Science of Bologna, Italy, <a href="https://www.spacetelescope.org/news/heic2016/" target="_blank" rel="noopener noreferrer">says</a> that "galaxy clusters are ideal laboratories in which to study whether the numerical simulations of the Universe that are currently available reproduce well what we can infer from gravitational lensing."</p>### Mapping dark matter

<span style="display:block;position:relative;padding-top:56.25%;" class="rm-shortcode" data-rm-shortcode-id="d904b585c806752f261e1215014691a6"><iframe type="lazy-iframe" data-runner-src="https://www.youtube.com/embed/fO0jO_a9uLA?rel=0" width="100%" height="auto" frameborder="0" scrolling="no" style="position:absolute;top:0;left:0;width:100%;height:100%;"></iframe></span><p>The assumption has been that the greater the lensing effect, the higher the concentration of dark matter.</p><p>As scientists analyzed the clusters' large-scale lensing — the massive arc and elongation visual effects produced by dark matter — they noticed areas of smaller-scale lensing within that larger distortion. The scientists interpret these as concentrations of dark matter within individual galaxies inside the clusters.</p><p>The researchers used spectrographic data from the VLT to determine the mass of these smaller lenses. <a href="https://www.oas.inaf.it/en/user/pietro.bergamini/" target="_blank" rel="noopener noreferrer">Pietro Bergamini</a> of the INAF-Observatory of Astrophysics and Space Science in Bologna, Italy explains, "The speed of the stars gave us an estimate of each individual galaxy's mass, including the amount of dark matter." The leader of the spectrographic aspect of the study was <a href="http://docente.unife.it/docenti-en/piero.rosati1/curriculum?set_language=en" target="_blank">Piero Rosati</a> of the Università degli Studi di Ferrara, Italy who recalls, "the data from Hubble and the VLT provided excellent synergy. We were able to associate the galaxies with each cluster and estimate their distances." </p><p>This work allowed the team to develop a thoroughly calibrated, high-resolution map of dark matter concentrations throughout the three clusters.</p>### But the models say...

<p>However, when the researchers compared their map to the concentrations of dark matter computer models predicted for galaxies bearing the same general characteristics, something was <em>way</em> off. Some small-scale areas of the map had 10 times the amount of lensing — and presumably 10 times the amount of dark matter — than the model predicted.</p><p>"The results of these analyses further demonstrate how observations and numerical simulations go hand in hand," notes one team member, <a href="https://nena12276.wixsite.com/elenarasia" target="_blank">Elena Rasia</a> of the INAF-Astronomical Observatory of Trieste, Italy. Another, <a href="http://adlibitum.oats.inaf.it/borgani/" target="_blank" rel="noopener noreferrer">Stefano Borgani</a> of the Università degli Studi di Trieste, Italy, adds that "with advanced cosmological simulations, we can match the quality of observations analyzed in our paper, permitting detailed comparisons like never before."</p><p>"We have done a lot of testing of the data in this study," Meneghetti says, "and we are sure that this mismatch indicates that some physical ingredient is missing either from the simulations or from our understanding of the nature of dark matter." <a href="https://physics.yale.edu/people/priyamvada-natarajan" target="_blank">Priyamvada Natarajan</a> of Yale University in Connecticut agrees: "There's a feature of the real Universe that we are simply not capturing in our current theoretical models."</p><p>Given that any theory in science lasts only until a better one comes along, Natarajan views the discrepancy as an opportunity, saying, "this could signal a gap in our current understanding of the nature of dark matter and its properties, as these exquisite data have permitted us to probe the detailed distribution of dark matter on the smallest scales."</p><p>At this point, it's unclear exactly what the conflict signifies. Do these smaller areas have unexpectedly high concentrations of dark matter? Or can dark matter, under certain currently unknown conditions, produce a tenfold increase in lensing beyond what we've been expecting, breaking the assumption that more lensing means more dark matter?</p><p>Obviously, the scientific community has barely begun to understand this mystery.</p>## A new minimoon is headed towards Earth, and it’s not natural

Astronomers spot an object heading into Earth orbit.

### Minimoons

<p>Scientists have confirmed just two prior minimoons. One was <a href="https://en.wikipedia.org/wiki/2006_RH120" target="_blank">2006 RH120</a>, which orbited us from September 2006 to June 2007. The other was <a href="https://en.wikipedia.org/wiki/2020_CD3" target="_blank">2020 CD3</a>, which got stuck in the 2015–2016 timeframe, and is believed to gotten away in May 2020.</p><p>2020 SO, the new kid on the block, is expected to arrive in October 2020 and pop out of orbit in May 2021.</p><div id="37962" class="rm-shortcode" data-rm-shortcode-id="f4c0fc8a2cba6536ea4cd960ebed3e6e"><blockquote class="twitter-tweet twitter-custom-tweet" data-twitter-tweet-id="1307729521869611008" data-partner="rebelmouse"><div style="margin:1em 0">Asteroid 2020 SO may get captured by Earth from Oct 2020 - May 2021. Current nominal trajectory shows shows capture… https://t.co/F5utxRvN6Z</div> — Tony Dunn (@Tony Dunn)<a href="https://twitter.com/tony873004/statuses/1307729521869611008">1600621989.0</a></blockquote></div>### Identifying 2020 SO

<p>The first clue 2020 SO isn't your ordinary asteroid is its exceptionally low velocity. It's traveling much more slowly that a typical asteroid — their <a href="https://www.lpi.usra.edu/exploration/training/illustrations/craterMechanics/" target="_blank">average rate of travel</a> <a href="https://www.lpi.usra.edu/exploration/training/illustrations/craterMechanics/" target="_blank" rel="noopener noreferrer"></a>is 18 kilometers (58,000 feet) per second. Even <a href="https://en.wikipedia.org/wiki/Moon_rock" target="_blank">moon rocks</a> sent careening into Earth orbit by impacts on the lunar surface outpace pokey 2020 SO.</p><p>For another thing, 2020 SO has an orbital path very similar to Earth's, lasting about one Earth year. It's also just slightly less circular than our own orbit, from which it's barely tilted off-axis.</p><p>So, what is it? <a href="https://cneos.jpl.nasa.gov/ca/" target="_blank">NASA estimates</a> that the object has dimensions very reminiscent of a discarded Centaur rocket stage from the <a href="https://en.wikipedia.org/wiki/Surveyor_2" target="_blank" rel="noopener noreferrer">Surveyor 2 mission</a> that landed an unmanned craft on the moon. Back in the day, rocket stages were jettisoned as craft were aimed toward their desired position. This stuff, if released high enough, remains in space. It appears that this Centaur rocket, launched in September 1966, is now making its way back homeward, at least for a little bit.</p><p>When 2020 SO arrives at its closest point in December, the rocket is expected to be about 50,000 kilometers from Earth. Its next closest approach is much further: 220,000 kilometers, in February 2010.</p><img type="lazy-image" data-runner-src="https://assets.rebelmouse.io/eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpbWFnZSI6Imh0dHBzOi8vYXNzZXRzLnJibC5tcy8yNDQzMDk3NC9vcmlnaW4uanBnIiwiZXhwaXJlc19hdCI6MTYyODg1MTQ1MX0.HGknDwqp0GmeuczKY_AS7vrPG7KMFUc_XO95tNoI2xo/img.jpg?width=980" id="e5cda" class="rm-shortcode" data-rm-shortcode-id="85eb1f790d8c3ee5b261f7ba13eaa5e1" data-rm-shortcode-name="rebelmouse-image" alt="Centaur rocket stage" />Centaur rocket stage

Credit: NASA/Wikimedia

### What we may be able to learn

<p>Earthly space programs being as young as they are, scientists would love to know what's happened to our rocket during a half century in space.</p><p>While 2020 SO won't get close enough to drop into our atmosphere, its slow progress has scientists hopeful that they'll still get some kind of a decent look at it.</p><p>Spectroscopy may be able to reveal what the rocket's surface is like now — has any of its paint survived, for example? Of course, being out in space, it's likely to have been hit by lots of dust and micrometeorites, so the current state of its surfaces is also of interest. Experts are curious to know how reflective the rocket is at this point, valuable information that can help planners of future long-term missions anticipate how well a craft out in space for extended periods will remain able to reflect sunlight.</p>## Did our early ancestors boil their food in hot springs?

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