That was the theory proposed in the 1860s by British physicist Lord Kelvin. While we now know that his theory is not correct, Kelvin's idea inspired the Scottish physicist and mathematician Peter Guthrie Tait to create a “table of the elements” by making a table of knots. Tait was not the only one tabulating knots. Reverend Thomas Kirkman, a British mathematician and minister, and Charles Little, an American mathematician, got in on the fun, too. And so began the rise of the mathematical field known as knot theory.

Unsurprisingly, mathematical knots are pretty similar to knots from everyday life. Suppose we take a long piece of rope or string, tie a knot in the middle, and then stick the ends together so that they are not loose. A mathematical knot is then "just such a knotted loop of string, except that we think of the string as having no thickness, its cross-section being a single point".

The "unknot" and the trefoil knot are the two simplest examples of mathematical knots. To make an unknot, simply take your piece of rope or string and glue its ends together, without tying a knot in the middle. To make the trefoil knot, first make an overhand knot and then glue its ends together.

We can represent both of these knots, and any other kind, by using a special kind of diagram called, well, a knot diagram. Think of these diagrams as capturing a knot’s shadow on the wall. For example, the image below shows a table of knots that have up to seven crossings, which are places where strands of the knot cross. At a crossing we leave a little bit of space in the diagram to indicate which strand goes underneath and which stays on top. The diagram for the unknot is on the top left and is drawn with zero crossings, while the diagram for the trefoil knot is next to it on the right and is drawn with three crossings.

An important question in knot theory is: how can we tell if two knots are the same? Intuitively, two knots are the same when we can turn one into the other by wiggling it around but without breaking it. Thanks to the German mathematician Kurt Reidemeister, we can show two knots are the same by working with knot diagrams. Reidemeister proved that two knots are the same if and *only* if we can turn a diagram of one into a diagram of the other by using a sequence of simple moves known as Reidemeister moves.

There are only three types of these moves. The first, called the Reidemeister type I move, allows you to add in or remove a twist to the diagram. The Reidemeister type II move allows you to add or take away two crossings. The third and final move, the Reidemeister type III move, allows you to move a strand in the knot diagram from one side of a crossing to the other.

The knot diagram below looks different to the diagram of the unknot that we saw earlier, but we can use Reidemeister moves to show that they represent the same knot. We just need to apply a type I Reidemeister move twice, to each of its twists, and we get back the familiar diagram of the unknot we saw earlier.

While that example was quite simple, in general trying to figure out if two knots are the same using Reidemeister moves is “pretty much impossible” says Jessica Purcell, professor of mathematics at Monash University. The problem, she explained via email, is that when trying to show two knots are the same “you can play around with Reidemeister moves and maybe you’ll magically move one into the other. But if you don’t, there’s no way to know if you can’t because the knots are different, or if you just haven’t found the right moves and need to keep trying.”

The difficulty of showing two knots are the same is illustrated by the story of the "Perko Pair" of knots. Charles Little thought that these knots were distinct and listed them as separate entries in one of his knot tables in 1899. John Conway checked Little’s table in the 1960s and didn’t notice anything suspicious about them. But in 1973 lawyer and mathematician Kenneth Perko "sketched a few diagrams to see if they might be the same" and found "a page and a half of yellow legal pad later" that they were. His short paper gave an abbreviated set of diagrams indicating how they could be transformed into each other. Perko’s discovery had important consequences by showing that a “theorem” Little thought he had proved was incorrect.

While it might seem shocking that the mistake about the Perko Pair went undetected for 74 years, Purcell doesn’t think it is too surprising. “After moving diagrams around for a long time, and finding no moves to turn one into the other, it makes sense that [early knot theorists] might give up and declare the knots to be different,” she said.

Now, however, mathematicians have a way to actually prove that two knots are different: tools called knot invariants. Imagine that we match up each knot with a label that has something written on it, such as a word or a number. This matching of knots to labels is called a "knot invariant" if equivalent knots, like the Perko pair, have the same label. To prove that two knots really are different, we need to find a type of knot invariant that matches those knots to labels that say different things.

As an example, consider a matching where a knot's label can say either "tricolorable" or "not tricolorable." A knot is assigned the "tricolorable" label when we can color each strand in its diagram one of three colors, say blue, red or green, while making sure that two conditions are met.

The first condition is that we must use at least two colors in the diagram (which, yes, is confusing when talking about "tri"colorability). The second is that, at each crossing, the three strands must either all be the same color or all be different colors. The trefoil gets assigned the "tricolorable" label because we can color it according to these rules, as depicted in the diagram.

On the other hand, if we cannot color a knot's diagram in this way then it gets assigned the "not tricolorable" label. The unknot, for example, is assigned the "not tricolorable" label because no matter how we try to color it, we can only use one color because it only has one arc to color, but we need to use at least two for a knot to be tricolorable. The matching of knots to the labels of "tricolorable" or "not tricolorable" is a knot invariant. So as we've seen that the trefoil is assigned "tricolorable" and the unknot is assigned "not tricolorable," this proves that they really are different knots.

Knot invariants are the topic of cutting edge research. Purcell explains that mathematicians find them “arising from many different places, including geometry, algebra and physics” and are trying to understand the connections between them.

She added, “I think some of the most exciting questions in knot theory right now involve how certain quantum invariants arising from physics relate to other information about knots arising from geometry.” In addition to physics, knot theory also has connections to biology and chemistry. DNA, for example, gets knotted and cut up into pieces by enzymes and knot theory can help scientists understand how this all works. In chemistry, knot theory can be applied to help scientists create knotted molecules. So while knot theory did not lead to a “table of the elements” like Lord Kelvin and Tait had hoped, it does have exciting links to physics, chemistry and biology.

]]>The researchers identified a substantial body of work by philosophers of science that used “philosophical tools to address scientific problems and provide scientifically useful proposals.” They call such work philosophy* in *science. So what kind of tools do philosophers use that can be applied to science? The study authors don’t offer an exhaustive list, but point to activities such as making distinctions and proposing definitions, critiquing scientific methods, and combining multiple scientific fields as examples of typical philosophical tools. And while scientists use these methods too, they don’t tend to do so as often or as rigorously as philosophers.

As an example, one philosophy in science paper referenced by the researchers is by philosophers Peter Heywood and Michael Redhead, published in *Foundations of Physics *in 1983. They prove that certain natural assumptions about the world are not compatible with quantum mechanics and, as Heywood and Redhead explicitly note, “an essential part” of their demonstration involves distinguishing between two different forms of an important concept in quantum mechanics that were often confused. So Heywood and Redhead address a scientific problem (are certain natural assumptions compatible with quantum mechanics?), use philosophical tools (making distinctions and proposing definitions) and provide a scientifically useful proposal (a proof).

The researchers suggest that, far from being fundamentally opposed disciplines, science and philosophy belong on a continuum. And when scientists and philosophers work together to combine their tools, they can make both scientific and philosophical advances. In the words of MC Hammer “It’s not science vs Philosophy ... It’s Science + Philosophy.”

]]>Most other mathematicians are not like Daubechies. As she notes, they tend to write in an “extremely terse” way that makes their work seem “opaque” to non-mathematicians. This is a big problem. Mathematics and science can influence each other, stimulating progress and leading to new discoveries. But if scientists cannot understand mathematicians then they won’t know about new mathematical tools they can use in their own work.

To try to fix this problem, we need to know why mathematicians tend to write in a way that is often inaccessible to scientists. In a recent paper discussing how to make math more accessible, Don Larson, Kristen Mazur, David White and Carolyn Yarnall point out that mathematicians are under pressure to write lots of research papers and publish them in prestigious journals. This can make mathematicians “feel compelled to write in a very specific style, aiming to present the results in the most concise language possible and to impress the reader with their brilliance.” Even if they try to adopt a more understandable style, mathematicians may encounter push back from reviewers or journals. Daubechies notes that she was lucky that the mathematics journal she published in “did not flinch” at the inclusion of her large table. Other journals may well have balked at it and, if they did, she would have had to remove it, making her paper less accessible to scientists.

The pressures mathematicians face when writing their research papers are unlikely to go away, so it may be difficult to make significant changes to the way they are written. Because of this, some mathematicians have been exploring alternative forms of writing to help make their work more accessible. One example is the User's Guide Project, a website where mathematicians can publish plain-language narratives about their research papers, making their technical work more user-friendly.

Participants in the project select one of their research papers and wrote a new kind of paper, called a user's guide, to accompany it. Each user's guide has sections containing background information, metaphors and diagrams, information about how the author came up with the results, as well as a summary aimed at non-mathematicians, with the explicit goal of making the original research paper understandable to a wider audience. The guides are peer-reviewed and made freely available on the project’s website. And while the user’s guides originally focused on an area of math called algebraic topology, in their paper, Larson, Mazur, White and Yarnall, who were all participants in the project, urge mathematicians from other areas to start their own version of it for their fields as well.

Despite the potential of user's guides to make math more accessible, Larson, Mazur, White and Yarnall report in their paper that many participants in the project did not receive credit for them from their institutions because the guides are not themselves research papers. This means that their institutions do not take the user’s guides into consideration when evaluating mathematicians for tenure and promotion.

This is not only unfair to the mathematicians who put in the time and effort to make their work more accessible, but will also discourage other mathematicians from participating in similar projects. After all, mathematicians want steady, secure employment like anyone else, so they will likely spend their time doing activities that their institutions recognize and reward, like writing new research papers, while avoiding the ones that they do not, like writing user’s guides.

As I argued recently, the lack of credit mathematicians receive for making their work easy to understand is something that can and should be changed. First, the mathematical community needs to be made more aware that making mathematics accessible can help researchers solve other scientific problems. Professional societies, such as the American Mathematical Society and the Mathematical Association of America, may be able to spread the word about this and potentially help mathematicians learn how to write these types of papers through outreach and professional development programs.

And if mathematicians are not sold on the importance of their work being used to solve scientific, rather than solely mathematical, problems, they should be reminded that new discoveries go both ways; scientific advances often bring with them mathematical advances, too. For example, Daubechies recounts how insights from engineering led to new mathematical results about wavelets, when an engineer named Martin Vetterli recognized a strong connection between wavelets and an engineering algorithm called subband filtering. And physicists found an important formula in an area of math called linear algebra while they were working out the behavior of subatomic particles.

Once the mathematical community is aware of the potential value of non-research papers like user’s guides, they must take action to formally acknowledge and give credit for them. One way to help this happen is to make sure that there are enough reputable journals that publish such work.

While there are a variety of journals that do publish these kinds of non-research papers, such as *Expositiones Mathematicae, *there has been discussion among some mathematicians that these types of journals are not well-known and there are not as many of them as there should be. One concrete fix for this would be to form a journal (or journals) that publishes user’s guides, as the papers generated by the User’s Guide Project were only published on the project’s own webpage. A second important way to acknowledge and reward work that makes mathematics more widely accessible is to fund projects that make this their goal. Professional societies may be able to help by lobbying funding agencies to make more money available for such projects.

Once work that makes mathematics more widely understandable is recognized and rewarded by the community, institutions will find it harder to ignore non-research papers like the user’s guides when evaluating mathematicians for tenure and promotion. And, hopefully, more mathematicians will be willing to write user’s guides and similar kinds of expository non-research papers, to connect new findings in mathematics to the scientists and other people who can apply them to real-world challenges.

As Daubechies said, "mathematicians who hope to collaborate with other scientists successfully must try to write their journal articles, and communicate in general, with their would-be collaborators in mind." We can encourage mathematicians to do this by making sure they receive fair credit for their work.

]]>Now, a new study by a group of Stanford University researchers published in *PLoS ONE* suggests that the attention retracted papers receive from news sites and social media is problematic as well. They found that popular articles, defined as those with an Altmetric attention score of more than 20, “receive substantially more attention than their retraction notice.” And it gets worse: The researchers found that the attention popular articles receive after they are retracted “does not always reflect their retraction, but may perpetuate” the flawed science they contain.

This suggests that journalists and social media users, as well as scientists, need to be wary of retracted papers. Fortunately, there are tools that can help with this. For scientists the reference manager Zotero will flag retracted papers and warn you before you cite them, and for journalists or members of the public who don’t use a reference manager, searching scite.ai for the title of the paper will let you know if it is problematic.

]]>But open access publishing comes with its own difficulties and obstacles. Some forms of open access require the author (or the author’s funder or institution) to pay a substantial fee, known as an article processing charge, so that the journal will publish their paper without a paywall. For example, in November of last year, the publisher *Nature* set the open-access fee for a subset of their journals at a staggering $11,390. Another obstacle is that fully open access journals don’t always have great reputations.

These obstacles to open-access publishing may hit researchers without much funding or job security especially hard. A recent study by a pair of scientists from the Academic Analytics Research Center analyzed which US researchers wrote open access papers from 2014 to 2018. They found that researchers who are male, have won federal grant funding, are affiliated with prestigious institutions, and/or work in STEM are more likely to have open access papers. In other words, the most privileged researchers in STEM disciplines are over-represented in the open access literature.

This means that important research by less privileged researchers is less likely to be freely available to all and, given the citation advantage of open access work, this means it is more likely to be ignored or overlooked. Despite its admirable intentions, the open access movement has work to do to ensure that open access literature better represents the full diversity of academia.

]]>In a recent article published in the journal *The Mathematics Intelligencer*, Cambridge mathematician Anthony Bordg compared this situation to the replication crisis in science. According to his analogy, attempts to check the correctness of a proof are like attempts to replicate a scientific experiment. If everything is correct, the proof has been replicated. But if an error is found or if mathematicians cannot determine whether the proof is correct for some other reason, then the attempt to replicate the proof has failed.

But too many proofs cannot be replicated. So, what can we do to fix this?

One potential solution is to use computers during the peer review process. Software called proof assistants can check if a proof is correct, once the proof has been translated into a language the software understands. While human reviewers do not always rigorously check that every single inference in a proof is correct, a proof assistant does. And if it finds anything amiss, it will complain. A proof assistant's blessing therefore counts as a successful replication of the proof “once and for all.”

But mathematicians have not yet welcomed proof assistants with open arms. One problem is that they are not exactly user friendly, and Bordg acknowledges that there is plenty of room for improvement. With further development, proof assistants will hopefully become more widely used in the near future.

]]>Unfortunately for Germain, mathematics was not regarded as a suitable subject for women in her time so she studied in secret, at night. When her parents discovered her night time study habit, they took away her fire, light, and even her clothes in an attempt to get her to stop studying and stay in bed. When even this failed, they relented. That did not mean she could study mathematics freely, however. Classes at the *École centrale des travaux publics*, later known as the *École Polytechnique*, were only available to men, but 18 year old Germain was able to obtain lecture notes for some of the classes. She then assumed the name of a male student, Monsieur LeBlanc, and wrote to one of the professors, Joseph-Louis Lagrange. Lagrange was impressed with Monsieur LeBlanc’s abilities and remained supportive when he found out LeBlanc was actually a woman.

Later, in 1804, Germain used the pseudonym Monsieur LeBlanc to write to another top mathematician: Carl Friedrich Gauss. Like Lagrange, Gauss was impressed with LeBlanc's abilities and they corresponded for a number of years. Germain eventually revealed her true identity to Gauss after Napoleon's 1806 invasion of Gauss's hometown. Once she learned of the invasion, Germain worried Gauss would meet the same fate as Archimedes and asked her friend General Pernety to protect him. Fortunately Gauss was safe, but when a French officer checked on him, explaining that a Parisian woman named Sophie Germain was concerned for his safety, he was confused as he did not know who she was. When this was reported back to Germain, she wrote to Gauss explaining the situation, revealing her true identity in the process. Fortunately, Gauss reacted positively to the news that LeBlanc was really Germain, telling her that women who overcome society’s prejudices to do mathematics have “the most noble courage, extraordinary talent, and superior genius."

During their correspondence, Germain and Gauss discussed their shared interest in number theory. Number theory is the branch of mathematics concerned with whole numbers and especially prime numbers. One of the most intriguing claims in number theory at the time was Fermat's Last Theorem: the equation x^{n}+ y^{n} = z^{n} , called the Fermat equation, has no positive whole number solutions for any whole number n>2 (so, for instance, 3^{2} + 4^{2} = 5^{2} works great, but no such equation exists for any exponent greater than two). It got its name from amateur mathematician Pierre de Fermat *claiming* he had a proof of it in around 1630, though he died without writing it down (and given what we know now it is very unlikely his "proof" was correct). Although Fermat’s Last Theorem is relatively simple to state, it wasn't proven for another two centuries after Germain and Gauss, in 1995 by Andrew Wiles with help from Richard Taylor.

Germain wrote to Gauss about Fermat’s Last Theorem on multiple occasions. In fact, she described some of her early attempts to prove it in her very first letter to him in 1804. In 1809, however, the French Academy of Sciences announced a prize competition to mathematically explain the behavior of vibrating surfaces, causing Germain to switch her focus to applied mathematics. She worked on this topic for a number of years and in 1816 was awarded the gold medal, making her the first woman to win a prize from the French Academy of Sciences. That same year, the Academy announced a new prize competition to prove Fermat’s Last Theorem and Germain began working on it again.

By the time the competition was announced, mathematicians had made some limited progress towards proving Fermat's Last Theorem. They could prove that it held for exponents n=3 and n=4, i.e. they could show that there were no positive whole number solutions to the equations x^{3} + y^{3} = z^{3} and x^{4} + y^{4} = z^{4}. They also knew that proving Fermat’s Last Theorem for all prime numbers greater than 2, i.e. showing that x^{p} + y^{p} = z^{p} has no solutions for p>2, was enough to prove Fermat’s Last Theorem in its entirety. To make further progress, however, they had to go beyond proving the result for specific exponents like 3 and 4. Germain did exactly this by proving, as mathematician and Germain biographer Dora Musielak describes it, "the first general result about arbitrary exponents for FLT [Fermat's Last Theorem]." This result is now known as Sophie Germain’s Theorem.

Sophie Germain's Theorem is quite technical, so let's focus on a simpler special case. First, there are Sophie Germain primes. A prime p is called a Sophie Germain prime if 2p+1 is also prime. So, for example, 3 is a Sophie Germain prime since 2 times 3 plus 1 is 7, which is also prime. 7, however, is not a Sophie Germain prime as 2 times 7 plus 1 is 15, which is not prime. As well as appearing in her work on Fermat's Last Theorem, Sophie Germain primes have important applications in cryptography, the mathematical study of codes and code-breaking.

The special case of Germain's Theorem says that for any Sophie Germain prime p>2, the equation x^{p} + y^{p} = z^{p} has no solutions when x times y times z is not divisible by p. This rules out a class of potential solutions to the Fermat equation when the exponent is a Sophie Germain prime. Germain also used the full version of her Theorem to show that for *any* prime p (not just Sophie Germain primes) greater than 2 and less than 100, the equation x^{p} + y^{p} = z^{p} has no solutions when x*y*z is not divisible by p.

Until around 2008, historians of mathematics thought that, while impressive and significant, Germain’s Theorem was her only contribution to Fermat’s Last Theorem. Then Andrea Del Centina and Reinhard Laubenbacher and David Pengelley examined her unpublished papers and discovered that Germain’s Theorem was just one piece of a much larger and sophisticated plan she had developed to prove all of Fermat’s Last Theorem. Her general idea was to show that any solutions to the equation x^{p} + y^{p} = z^{p} must have infinitely many prime divisors. This would mean that there cannot be any solutions, because any number divisible by infinitely many primes must itself be infinite. Unfortunately, her plan could not be made to work, as she herself came to recognize. Nonetheless, it was still fruitful. In her pursuit of it, Germain developed new methods and proved new results that were later rediscovered by others.

Despite Germain's brilliance, the continual obstacles she faced due to her gender made it impossible for her to become a professional mathematician. She thus remained an amateur throughout her life, never obtaining a position at a university. Her mathematical work, however, was sophisticated and exemplified her boldness and creativity, just like her earlier efforts to overcome barriers to gain a mathematical education.

Musielak suggested in an email interview that the obstacles Germain faced may have even shaped her approach to Fermat's Last Theorem: "Maybe because she was an amateur mathematician, determined to arrive at a proof, working alone with all odds against her, Germain had to think differently. In the end, Sophie Germain developed her own algorithms and a unique approach to prove Fermat's theorem, distinctively different from Euler’s and Fermat's."

]]>The Four Color Theorem states that for any map of contiguous countries drawn on a plane only four colors are needed to ensure that adjacent countries are given different colors. In the heart of his “proof,” Kempe had to consider a number of different cases of possible map configurations. To tackle these cases, he invented a new mathematical tool called Kempe chains. However Heawood discovered that Kempe's treatment of one of these cases didn’t work and presented an example to show how his reasoning failed.

Heawood's discovery did not mean that all was lost, though. First, Heawood showed that Kempe's work was enough to establish a weaker result called the Five Color Theorem which says that only five colors are needed to color a map in the required way. Second, although Kempe's proof was flawed, the Four Color Theorem was true. It was first proved successfully, though somewhat controversially, using a computer by Kenneth Appel and Wolfgang Haken nearly a century later in 1976. And Kempe’s ideas played a significant role in their work. So while Kempe made a mistake in his “proof,” it still contained valuable mathematics.

]]>MC Hammer's insistence on the complementary nature of science and philosophy is in line with this 2019 opinion paper, published in *PNAS*. The authors described a continuum of science and philosophy, as the two fields share "the tools of logic, conceptual analysis, and rigorous argumentation.” They also provided three concrete examples of how philosophy helped scientific research in the life sciences and concluded with six practical proposals to encourage collaboration between scientists and philosophers.

Tweets by MC Hammer promoting these views will hopefully also help to break down harmful stereotypes of the disciplines that might prevent scientists and philosophers from working together for the good of society.

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