Author: IMC

A mathematical link between two key equations—one that deals with the very big and the other, the very small—has been developed by a young mathematician in China.

The mathematical discipline known as differential geometry is concerned with the geometry of smooth shapes and spaces. With roots going back to antiquity, the field flourished in the early 20th century, enabling Einstein to develop his general theory of relativity and other physicists to develop quantum field theory and the Standard Model of particle physics.

Gao Chen, a 29-year-old mathematician at the University of Science and Technology of China in Hefei, specializes in a branch known as complex differential geometry. Its complexity is not in dealing with complicated structures, but rather because it is based on complex numbers—a system of numbers that extends everyday numbers by including the square root of -1.

This area appeals to Chen because of its connections with other fields. “Complex differential geometry lies at the intersection of analysis, algebra, and mathematical physics,” he says. “Many tools can be used to study this area.”

Chen has now found a new link between two important equations in the field: the Kähler–Einstein equation, which describes how mass causes curvature in space–time in general relativity, and the Hermitian–Yang–Mills equation, which underpins the Standard Model of particle physics.

Chen was inspired by his Ph.D. supervisor Xiuxiong Chen of New York’s Stony Brook University, to take on the problem. “Finding solutions to the Hermitian–Yang–Mills and the Kähler–Einstein equations are considered the most important advances in complex differential geometry in previous decades,” says Gao Chen. “My results provide a connection between these two key results.”

“The Kähler –Einstein equation describes very large things, as large as the universe, whereas the Hermitian–Yang–Mills equation describes tiny things, as small as quantum phenomena,” explains Gao Chen. “I’ve built a bridge between these two equations.” Gao Chen notes that other bridges existed previously, but that he has found a new one.

“This bridge provides a new key, a new tool for theoretical research in this field,” Gao Chen adds. His paper describing this bridge was published in the journal Inventiones mathematicae in 2021.

In particular, the finding could find use in string theory—the leading contender of theories that researchers are developing in their quest to unite quantum physics and relativity. “The deformed Hermitian–Yang–Mills equation that I studied plays an important role in the study of string theory,” notes Gao Chen.

Gao Chen now has his eyes set on other important problems, including one of the seven Millennium Prize Problems. These are considered the most challenging in the field by mathematicians and carry a $1 million prize for a correct solution. “In the future, I hope to tackle a generalization of the Kähler–Einstein equation,” he says. “I also hope to work on other Millennium Prize problems, including the Hodge conjecture.”

Provided by University of Science and Technology of China

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Credit of the article given to University of Science and Technology of China

Proofs, the central tenet of mathematics, occasionally have errors in them. Could computers stop this from happening, asks mathematician Emily Riehl.

Computer proof assistants can verify that mathematical proofs are correct

One miserable morning in 2017, in the third year of my tenure-track job as a mathematics professor, I woke up to a worrying email. It was from a colleague and he questioned the proof of a key theorem in a highly cited paper I had co-authored. “I had always kind of assumed that this was probably not true in general, though I have no proof either way. Did I miss something?” he asked. The proof, he noted, appeared to rest on a tacit assumption that was not warranted.

Much to my alarm and embarrassment, I realised immediately that my colleague was correct. After an anxious week working to get to the bottom of my mistake, it turned out I was very lucky. The theorem was true; it just needed a new proof, which my co-authors and I supplied in a follow-up paper. But if the theorem had been false, the whole edifice of consequences “proven” using it would have come crashing down.

The essence of mathematics is the concept of proof: a combination of assumed axioms and logical inferences that demonstrate the truth of a mathematical statement. Other mathematicians can then attempt to follow the argument for themselves to identify any holes or convince themselves that the statement is indeed true. Patched up in this way, theorems originally proven by the ancient Greeks about the infinitude of primes or the geometry of planar triangles remain true today – and anyone can see the arguments for why this must be.

Proofs have meant that mathematics has largely avoided the replication crises pervading other sciences, where the results of landmark studies have not held up when the experiments were conducted again. But as my experience shows, mistakes in the literature still occur. Ideally, a false claim, like the one I made, would be caught by the peer review process, where a submitted paper is sent to an expert to “referee”. In practice, however, the peer review process in mathematics is less than perfect – not just because experts can make mistakes themselves, but also because they often do not check every step in a proof.

This is not laziness: theorems at the frontiers of mathematics can be dauntingly technical, so much so that it can take years or even decades to confirm the validity of a proof. The mathematician Vladimir Voevodsky, who received a Fields medal, the discipline’s highest honour, noted that “a technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail”. After several experiences in which mistakes in his proofs took over a decade to be resolved – a long time for something to sit in logical limbo – Voevodsky’s subsequent crisis of confidence led him to take the unusual step of abandoning his “curiosity-driven research” to develop a computer program that could verify the correctness of his work.

This kind of computer program is known as a proof assistant, though it might be better called a “proof checker”. It can verify that a string of text proves the stated theorem. The proof assistant knows the methods of logical reasoning and is equipped with a library of proofs of standard results. It will accept a proof only after satisfying each step in the reasoning process, with no shortcuts of the sort that human experts often use.

For instance, a computer can verify that there are infinitely many prime numbers by validating the following proof, which is an adaptation of Greek mathematician Euclid’s argument. The human mathematician first tells the computer exactly what is being claimed – in this case that for any natural number N there is always some prime number p that is larger. The human then tells the computer the formula, defining p to be the minimum prime factor of the number formed by multiplying all the natural numbers up to N together and adding 1, represented as N! + 1.

For the computer proof assistant to make sense of this, it needs a library that contains definitions of the basic arithmetic operations. It also needs proofs of theorems, like the fundamental theorem of arithmetic, which tells us that every natural number can be factored uniquely into a product of primes. The proof assistant then demands a proof that this prime number p is greater than N. This is argued by contradiction – a technique where following an assumption to its conclusion leads to something that cannot possibly be true, demonstrating that the original assumption was false. In this case, if p is less than or equal to N, it should be a factor of both N! + 1 and N!. Some simple mathematics says this means that p must also be a factor of 1, which is absurd.

Computer proof assistants can be used to verify proofs that are so long that human referees are unable to check every step. In 1998, for example, Samuel Ferguson and Thomas Hales announced a proof of Johannes Kepler’s 1611 conjecture that the most efficient way to pack spheres into three-dimensional space is the familiar “cannonball” packing. When their result was accepted for publication in 2005 it came with a caveat: the journal’s reviewers attested to “a strong degree of conviction of the essential correctness of this proof approach” – they declined to certify that every step was correct.

Ferguson and Hales’s proof was based on a strategy proposed by László Fejes Tóth in 1953, which reduced the Kepler conjecture to an optimisation problem in a finite number of variables. Ferguson and Hales figured out how to subdivide this optimisation problem into a few thousand cases that could be solved by linear programming, which explains why human referees felt unable to vouch for the correctness of each calculation. In frustration, Hales launched a formalisation project, where a team of mathematicians and computer scientists meticulously verified every logical and computational step in the argument. The resulting 22-author paper was published in 2017 to as much fanfare as the original proof announcement.

Computer proof assistants can also be used to verify results in subfields that are so technical that only specialists understand the meaning of the central concepts. Fields medallist Peter Scholze spent a year working out the proof of a theorem that he wasn’t quite sure he believed and doubted anyone else would have the stamina to check. To be sure that his reasoning was correct before building further mathematics on a shaky foundation, Scholze posed a formalisation challenge in a SaiBlog post entitled the “liquid tensor experiment” in December 2020. The mathematics involved was so cutting edge that it took 60,000 lines of code to formalise the last five lines of the proof – and all the background results that those arguments relied upon – but nevertheless this project was completed and the proof confirmed this past July by a team led by Johan Commelin.

Could computers just write the proofs themselves, without involving any human mathematicians? At present, large language models like ChatGPT can fluently generate mathematical prose and even output it in LaTeX, a typesetting program for mathematical writing. However, the logic of these “proofs” tends to be nonsense. Researchers at Google and elsewhere are looking to pair large language models with automatically generated formalised proofs to guarantee the correctness of the mathematical arguments, though initial efforts are hampered by sparse training sets – libraries of formalised proofs are much smaller than the collective mathematical output. But while machine capabilities are relatively limited today, auto-formalised maths is surely on its way.

In thinking about how the human mathematics community might wish to collaborate with computers in the future, we should return to the question of what a proof is for. It’s never been solely about separating true statements from false ones, but about understanding why the mathematical world is the way it is. While computers will undoubtedly help humans check their work and learn to think more clearly – it’s a much more exacting task to explain mathematics to a computer than it is to explain it to a kindergartener – understanding what to make of it all will always remain a fundamentally human endeavour.

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*Credit for article given to Emily Riehl*

Imagine if you enrolled your child in swimming lessons but instead of a qualified swimming instructor, they were taught freestyle technique by a soccer coach.

Something similar is happening in classrooms around Australia every day. As part of the ongoing teacher shortage, there are significant numbers of teachers teaching “out-of-field”. This means they are teaching subjects they are not qualified to teach.

One of the subjects where out-of-field teaching is particularly common is maths.

A 2021 report on Australia’s teaching workforce found that 40% of those teaching high school mathematics are out-of-field (English and science were 28% and 29%, respectively).

Another 2021 study of students in Year 8 found they were more likely to be taught by teachers who had specialist training in both maths and maths education if they went to a school in an affluent area rather than a disadvantaged one (54% compared with 31%).

Our new report looks at how we can fix this situation by training more existing teachers in maths education.

Why is this a problem?

Mathematics is one of the key parts of school education. But we are seeing worrying signs students are not receiving the maths education they need.

The 2021 study of Year 8 students showed those taught by teachers with a university degree majoring in maths had markedly higher results, compared with those taught by out-of-field teachers.

We also know maths skills are desperately needed in the broader workforce. The burgeoning worlds of big data and artificial intelligence rely on mathematical and statistical thinking, formulae and algorithms. Maths has also been identified as a national skill shortages priority area.

There are worrying signs students are not receiving the maths education they need. Aaron Lefler/ Unsplash, CC BY

What do we do about this?

There have been repeated efforts to address teacher shortages,including trying to retain existing mathematics teachers, having specialist teachers teaching across multiple schools and higher salaries. There is also a push to train more teachers from scratch, which of course will take many years to implement.

There is one strategy, however, that has not yet been given much attention by policy makers: upgrading current teachers’ maths and statistics knowledge and their skills in how to teach these subjects.

They already have training and expertise in how to teach and a commitment to the profession. Specific training in maths will mean they can move from being out-of-field to “in-field”.

How to give teachers this training

A new report commissioned by mathematics and statistics organisations in Australia (including the Australian Mathematical Sciences Institute) looks at what is currently available in Australia to train teachers in maths.

It identified 12 different courses to give existing teachers maths teaching skills. They varied in terms of location, duration (from six months to 18 months full-time) and aims.

For example, some were only targeted at teachers who want to teach maths in the junior and middle years of high school. Some taught university-level maths and others taught school-level maths. Some had government funding support; others could cost students more than A$37,000.

Overall, we found the current system is confusing for teachers to navigate. There are complex differences between states about what qualifies a teacher to be “in-field” for a subject area.

In the current incentive environment, we found these courses cater to a very small number of teachers. For example, in 2024 in New South Wales this year there are only about 50 government-sponsored places available.

This is not adequate. Pre-COVID, it was estimated we were losing more than 1,000 equivalent full-time maths teachers per year to attrition and retirement and new graduates were at best in the low hundreds.

But we don’t know exactly how many extra teachers need to be trained in maths. One of the key recommendations of the report is for accurate national data of every teacher’s content specialisations.

We need a national approach

The report also recommends a national strategy to train more existing teachers to be maths teachers. This would replace the current piecemeal approach.

It would involve a standard training regime across Australia with government and school-system incentives for people to take up extra training in maths.

There is international evidence to show a major upskilling program like this could work.

In Ireland, where the same problem was identified, the government funds a scheme run by a group of universities. Since 2012, teachers have been able to get a formal qualification (a professional diploma). Between 2009 and 2018 the percentage of out-of-field maths teaching in Ireland dropped from 48% to 25%.

To develop a similar scheme here in Australia, we would need coordination between federal and state governments and universities. Based on the Irish experience, it would also require several million dollars in funding.

But with students receiving crucial maths lessons every day by teachers who are not trained to teach maths, the need is urgent.

The report mentioned in this article was commissioned by the Australian Mathematical Sciences Institute, the Australian Mathematical Society, the Statistical Society of Australia, the Mathematics Education Research Group of Australasia and the Actuaries Institute.

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Credit of the article given to Monstera Production/ Pexels , CC BY

Life, from the perspective of thermodynamics, is a system out of equilibrium, resisting tendencies towards increasing their levels of disorder. In such a state, the dynamics are irreversible over time. This link between the tendency toward disorder and irreversibility is expressed as the ‘arrow of time’ by the English physicist Arthur Eddington in 1927.

Now, an international team including researchers from Kyoto University, Hokkaido University, and the Basque Center for Applied Mathematics, has developed a solution for temporal asymmetry, furthering our understanding of the behaviour of biological systems, machine learning, and AI tools.

“The study offers, for the first time, an exact mathematical solution of the temporal asymmetry—also known as entropy production—of nonequilibrium disordered Ising networks,” says co-author Miguel Aguilera of the Basque Center for Applied Mathematics.

The researchers focused on a prototype of large-scale complex networks called the Ising model, a tool used to study recurrently connected neurons. When connections between neurons are symmetric, the Ising model is in a state of equilibrium and presents complex disordered states called spin glasses. The mathematical solution of this state led to the award of the 2021 Nobel Prize in physics to Giorgio Parisi.

Unlike in living systems, however, spin crystals are in equilibrium and their dynamics are time reversible. The researchers instead worked on the time-irreversible Ising dynamics caused by asymmetric connections between neurons.

The exact solutions obtained serve as benchmarks for developing approximate methods for learning artificial neural networks. The development of learning methods used in multiple phases may advance machine learning studies.

“The Ising model underpins recent advances in deep learning and generative artificial neural networks. So, understanding its behaviour offers critical insights into both biological and artificial intelligence in general,” added Hideaki Shimazaki at KyotoU’s Graduate School of Informatics.

“Our findings are the result of an exciting collaboration involving insights from physics, neuroscience and mathematical modeling,” remarked Aguilera. “The multidisciplinary approach has opened the door to novel ways to understand the organization of large-scale complex networks and perhaps decipher the thermodynamic arrow of time.”

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Credit of the article given to Kyoto University

It’s a dilemma that many a regular bettor probably faces often—deciding when to place a sports bet. In a study entitled, “A statistical theory of optimal decision-making in sports betting,” Jacek Dmochowski, Associate Professor in the Grove School of Engineering at The City College of New York, provides the answer. His original finding appears in the journal PLOS One.

“The central finding of the work is that the objective in sports betting is to estimate the median outcome. Importantly, this is not the same as the average outcome,” said Dmochowski, whose expertise includes machine learning, signal processing and brain-computer interfaces. “I approach this from a statistical point-of-view, but also provide some intuitive results with sample data from the NFL that can be digested by those without a background in math.”

To illustrate one of the findings, he presents a hypothetical example. “Assume that Kansas City has played Philadelphia three times previously. Kansas City has won each of those games by margins of 3, 7, and 35 points. They are playing again, and the point spread has been posted as ‘Kansas City -10.’ This means that Kansas City is favoured to win the game by 10 points according to the sportsbooks.”

For a bettor, Dmochowski added, the optimal decision in this scenario is to bet on Philadelphia (+10), even though they have lost the last three games by an average margin of 15 points. The reason is that the median margin of victory in those games was only 7, which is less than the point spread of 10.

He noted that because a bettor’s intuition may sometimes be more linked to an average outcome rather than the median, the utilization of some data, or even better, a model, is strongly encouraged.

On his new findings, Dmochowski said he was surprised that the derived theorems have not been previously presented, although it is possible that sports books and some statistically-minded bettors have understood at least the basic intuitions that are conveyed by the math.

Moreover, other investigators have reported findings that align with what’s in the paper, principally Fabian Wunderlich and Daniel Memmert at the German Sports University of Cologne.

With a Pew Research poll establishing that one in five Americans have placed a sports bet in the last year, Dmochowski’s study should be of interest to many bettors in this growing enterprise.

He had other advice for potential bettors. Firstly, “Avoid betting on matches for which the sports book has produced estimates that are ‘very close’ to the median outcome. In the case of the National Football League, the analysis shows that ‘very close’ is equivalent to the point spread being within one point of the true median.”

“Secondly, understand that the sports books are incredibly skilled at setting the odds. At the same time, they only need to make a small error to allow a profitable bet. So the goal is to seek out those opportunities.”

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Credit of the article given to Jay Mwamba, City College of New York

Whether it’s physical phenomena, share prices or climate models—many dynamic processes in our world can be described mathematically with the aid of partial differential equations. Thanks to stochastics—an area of mathematics which deals with probabilities—this is even possible when randomness plays a role in these processes.

Something researchers have been working on for some decades now are so-called stochastic partial differential equations. Working together with other researchers, Dr. Markus Tempelmayr at the Cluster of Excellence Mathematics Münster at the University of Münster has found a method which helps to solve a certain class of such equations.

The results have been published in the journal Inventiones mathematicae.

The basis for their work is a theory by Prof. Martin Hairer, recipient of the Fields Medal, developed in 2014 with international colleagues. It is seen as a great breakthrough in the research field of singular stochastic partial differential equations. “Up to then,” Tempelmayr explains, “it was something of a mystery how to solve these equations. The new theory has provided a complete ‘toolbox,’ so to speak, on how such equations can be tackled.”

The problem, Tempelmayr continues, is that the theory is relatively complex, with the result that applying the ‘toolbox’ and adapting it to other situations is sometimes difficult.

“So, in our work, we looked at aspects of the ‘toolbox’ from a different perspective and found and proved a method which can be used more easily and flexibly.”

The study, in which Tempelmayr was involved as a doctoral student under Prof. Felix Otto at the Max Planck Institute for Mathematics in the Sciences, published in 2021 as a pre-print. Since then, several research groups have successfully applied this alternative approach in their research work.

Stochastic partial differential equations can be used to model a wide range of dynamic processes, for example, the surface growth of bacteria, the evolution of thin liquid films, or interacting particle models in magnetism. However, these concrete areas of application play no role in basic research in mathematics as, irrespective of them, it is always the same class of equations which is involved.

The mathematicians are concentrating on solving the equations in spite of the stochastic terms and the resulting challenges such as overlapping frequencies which lead to resonances.

Various techniques are used for this purpose. In Hairer’s theory, methods are used which result in illustrative tree diagrams. “Here, tools are applied from the fields of stochastic analysis, algebra and combinatorics,” explains Tempelmayr. He and his colleagues selected, rather, an analytical approach. What interests them in particular is the question of how the solution of the equation changes if the underlying stochastic process is changed slightly.

The approach they took was not to tackle the solution of complicated stochastic partial differential equations directly, but, instead, to solve many different simpler equations and prove certain statements about them.

“The solutions of the simple equations can then be combined—simply added up, so to speak—to arrive at a solution for the complicated equation which we’re actually interested in.” This knowledge is something which is used by other research groups who themselves work with other methods.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Kathrin Kottke, University of Münster

In teaching mathematics, the challenge often lies in making abstract concepts and problem-solving strategies explicit to students. Students often struggle to grasp the underlying reasoning and processes that drive mathematics. Explicit Teaching in Mathsprovides strategies for the teacher which help them to design and present instruction and learning to students in a meaningful way.

Teacher modelling is a way we can intentionally make clear the reasoning and processes behind mathematical ideas and concepts. Listen to our podcast, Explicit modelling of reasoning and processes behind actions , for more, where Allan Dougan (AAMT) and Dr Kristen Trippet (Australian Academy of Science) discuss how we model the reasoning and processes behind actions.

This article answers the questions:

• What do we mean by modelling in maths?
• How do we model the reasoning and processes behind actions in maths?
• How do we make the most of modelling in classroom teaching?

What is Teacher modelling in mathematics?

Mathematics is a subject where concepts can easily remain invisible to students. Teacher modelling allows teachers to showcase and explain the reasoning and processes behind mathematical concepts and practices. It often involves breaking down complex problems, strategies and skills into manageable steps that students can understand and replicate.

Teacher modelling is about thinking aloud and making visible those practices that are often concealed within a mathematician’s mind.

What is mathematical reasoning?

According to the Australian Curriculum V9.0: Understand this learning area , students are reasoning mathematically when they can:

• explain their thinking
• deduce and justify strategies used and conclusions reached
• adapt the known to the unknown
• transfer learning from one context to another
• prove that something is true or false.

Explicit modelling is a powerful tool for developing students’ mathematical reasoning abilities. Model mathematical reasoning by analysing, generalising and justifying mathematical situations. By thinking aloud, teachers can model this process effectively.

Start by analysing a problem: 7+8, I can do that with 5+2+8!

Then generalise the approach: It doesn’t matter what order I add these numbers up in, I will get the same result.

Justify the learning: I can use this approach to solve other problems too.

Using mathematical reasoning helps build patterns of thinking and mathematical fluency.

Beyond worked examples

Worked examples are the prime example of modelling and mathematical reasoning in action, but they are just the tip of the iceberg. While they provide a helpful starting point for students, relying solely on worked examples will not lead to a comprehensive understanding of mathematical concepts.

Fade in/Fade out

The idea of Fade in/Fade out support refers to when and why support is provided. This concept is useful when thinking about modelling.

Teachers will often start a lesson with a worked example and then fade out their modelling while students proceed with their learning. This approach doesn’t allow for the flexibility and responsiveness required for the most effective modelling. Modelling should not be a one-time occurrence. Instead, it should be scaffolded throughout a lesson. Build your modelling up through your lesson and fade in with instruction and modelling as students need it.

What does explicit modelling and mathematical reasoning look like?

Let’s unpack it with an example.

Example 1:

Problem: A student is asked to solve the equation 7+8 and to explain how they did it.

Student: I added 2 to 8, which made 10, and then added 5.

This strategy is called ‘bridging to 10’ and involves children using their knowledge of addition up to 10 as a base to then work out sums with totals over 10. A modelled response to this student’s work might look like this:

Teacher: Hang on a second, you said you added 2 to the 8, let’s take a look at that. So, 7+8 is the same as 5+2+8, where did the 7 go in that? Oh yes! (5+2)+8 also equals 15! What happens if we change the order of the numbers?

Student: 8+5+2, that also equals 15.

In this example, the student has modelled their own mathematical awareness, and this has then been extended with the teacher modelling both equivalence and the associative property in the working out of the problem.

Planning for modelling

Modelling is most effective when it is intentional and focused. Before teaching a lesson, ask yourself:

• What is the mathematical concept I want students to understand?
• What is the important mathematical skill I want them to see?
• What are the most effective ways I can model these strategies to students?
• How can I be responsive in my modelling (for example, fading in)?

There is so much more to maths than just getting the right answer! Explicit modelling is a powerful tool for making mathematical concepts and practices accessible to students. By incorporating modelling, educators empower students to not only find the right answers, but also to develop their mathematical reasoning skills, as well as their understanding of the underlying processes and strategies that mathematics is built upon. Modelling helps students to develop their thinking and come up with efficient strategies for tackling unknown mathematical problems. And this is the work of a true mathematician!

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Credit of the article given to The Mathematics Hub

If mathematics is a game, then playing some game is doing mathematics, and in that case why isn’t dancing mathematics too?

Ludwig Wittgenstein – Remarks on the Foundations of Mathematics

Mathematics is often described metaphorically – the  forms that these metaphors take include the organic, mechanical, classical, and post-modern, among countless others. Within these metaphors, mathematics may be a tool, or set of tools, a tree, part of a tree, a vine, a game, or set of games, and mathematicians in turn may be machines, game-players, artists, inventors, or explorers.

Despite the many metaphors used to describe mathematics, in popular discourse mathematics is often reduced to one of its parts, being metonymically described as merely about numbers, formulas, or some other limited aspect. Metaphor is a more complete substitution of ideas than metonymy – allowing us to link concepts that do not appear to have any direct relationship. Perhaps, metaphoric language that elevates and expands our ideas about mathematics is used by enthusiasts to counter the more limited and diminishing metonymic descriptions that are often encountered.

Attempts to describe and elevate mathematics through metaphor seem to fall short, however. Our usual way of thinking about things is to inquire about their meaning – a meaning that is assumed to lie beneath or beyond mere appearances. Metaphor generally relies on making connections between concepts on this deeper level. The sheer formalism of mathematics frustrates this usual way of thinking, and leaves us grasping for a meaning that is constantly evasive. The sheer number and variety of the  many metaphors for mathematics suggests that no single convincing one has yet been found. It may be that the repeated attempts to find such a unifying metaphor represents an ongoing and forever failing attempt to grapple with the purely formal character of mathematics; and it may be that the formal nature of mathematics will always shake off any metaphor that attempts to tie it down.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to dan.mackinnon*

In many traditions, Biedermann’s Dictionary of Symbols tells us, “the tree was widely seen as the axis mundi around which the cosmos is organized” and, as mentioned in a previous post, has been widely used to describe the relationship between mathematics and other sciences. Mathematics itself, like many subjects, is often portrayed as a tree whose sub-topics make up branches that continue to grow and bifurcate.

Some recent articles have take a more postmodern perspective on using the tree metaphor to describe mathematics.

Dan Kennedy ‘s “Climbing around on the Tree of Mathematics,” (full text here) and Greg McColm’s “A Metaphor for Mathematics Education” are two recent articles that make arguments by analogy about what mathematics is and how it should be taught. In Kennedy’s argument, mathematics is a tree, while in McColm’s it is a vine – both are organic, growing, and branching. What distinguishes these two uses of metaphor from traditional tree analogies is that both authors are not at all suggesting that we can stand back and survey the structure as a whole and understand how all its parts are related. The ability to provide a comprehensive view of the subject, to make it surveyable, was the raison-d’être of metaphors like the “Tree of Science.” Instead of using the metaphor this way, both authors suggest that we think of ourselves as part of the growing structure – as climbers and gardeners who cannot see the complex organic whole, but who can explore and tend to our small part of it. In these descriptions, natural forms like trees and plants, once metaphors for simplicity and comprehensibility, now provide metaphors for complexity.

Up in the Tree of Mathematics, Kennedy suggests that working mathematicians are labouring at extending its outer branches. This is where the view is best, where the fruit is found, and where the beauty of mathematics can be seen most clearly. School Mathematics is part of the trunk, the solid, oldest, stable part of the tree, and math teachers spend their time helping students climb the trunk, hoping that some may one day reach its outer branches. Unfortunately, the difficulty of the trunk prevents most people from ever climbing beyond it. Kennedy suggests that we should be less concerned with the trunk than with the branches, and that technology can provide a ladder to assist the climb.

McColm’s Mathematical Vine is not mathematics itself, but a structure that clings to the underlying reality of mathematical truth. Mathematics, in this analogy, is like a hidden tower, whose shape can only be seen by looking at the vine that has taken shape around it. Like in Kennedy’s analogy, working mathematicians are the caretakers who help the structure grow. For McColm, this analogy emphasizes the importance of mathematics education – a process of strengthening the vine so that it may continue to grow. Perhaps because his audience is primarily post-secondary researchers, he does not advocate finding shortcuts to “higher” views, but rather suggests that education be promoted through “tending to the vine” – clarifying mathematics and strengthening connections between different branches.

Although they suggest more of a structure at play, rather that a stable unified whole, organic metaphors like those used by McColm and Kennedy continue to suggest a natural unity among the various parts of mathematics. In that sense they are still rooted (or centered), and, although they have somewhat destabilized the tree analogy, they haven’t quite deconstructed it. They have not, for example, gone quite as far as Wittgenstein, who seemed to suggest that metaphors that attempt to link the subjects of mathematics in a defining way like this are misguided. In his view, as described by Ackerman (1988, p. 115):

mathematics is an assemblage of language games, having no sharp and uniform external boundary, with potentially confusing and criss-crossing subdisciplines held together by an internal network of analogous proof techniques.

It is easy to appreciate how some climbers in Kennedy’s trees and McColm’s vines end up like the protagonist in Roz Chast’s cartoon “Falling off the Math Cliff”, where step 1 is “A boy begins his wondrous journey,” and step 8 is “The plummet.”

The images in this post are “Pythagoras Tree” fractals, made using GSP.

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*Credit for article given to dan.mackinnon*

Robert Recorde, author the first English textbook on algebra (published in 1557), chose to give his book the metaphorical title The Whetstone of Witte to encourage people to take up the new and difficult practice of algebra. The metaphor of a whetstone, or blade-sharpener, suggests that algebra is not only useful, but also good mental exercise. In the verse that he included on its title page, he writes,

Its use is great, and more than one. Here if you lift your wits to wet, Much sharpness thereby shall you get. Dull wits hereby do greatly mend, Sharp wits are fined to their full end.

Mathematics, and algebra in particular, according to The Whetstone of Witte is like a knife-sharpener for the brain. Four hundred years later, in his book Mathematician’s Delight (1961), W.W. Sawyer takes up a similar metaphor, suggesting that “Mathematics is like a chest of tools: before studying the tools in detail, a good workman should know the object of each, when it is used, how it is used.” Whether they describe mathematics as a sharpener or other tool, these mechanical metaphors are commonly used to emphasize the practicality and versatility of mathematics, particularly when employed in engineering or science, and suggest that it should be used thoughtfully, and with precision.

An often quoted mechanical metaphor that suggests a more frantic and less precise process of mathematical creation is often attributed to Paul Erdos: “a mathematician is a machine for turning coffee into theorems.”

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*Credit for article given to dan.mackinnon*