Irix is a company that combines
exact mathematical data,
complex optimisation and
metamathematics to create
a platform for general
mathematical insight.
It helps users discover structure,
raise new conjectures and
link disparate problems
across science and industry.
How we obtain mathematical insight has hardly changed since Newton invented calculus and the laws of motion 350 years ago. No other domain of human endeavour has been more resistant to technological improvement.
For centuries, mathematical insight has relied on paper and pencil to work out examples and manipulate formulae. Only two technological advances have had an impact. First, computers made it easier to work out examples and collect data. Second, symbolic computation software, such as Mathematica, expedited manipulating and solving equations. Together, these advances sped up mathematical discovery by perhaps 50%—a modest gain compared to the giant leaps in efficiency technology has delivered elsewhere. Achieving a step change in mathematical insight requires a radically new approach.
AN OPEN PROBLEM
When it comes to reducing toil, improving health and boosting security, artificial intelligence has been transformative. From text translation to medical imaging to autonomous flight, AI helps us solve problems better and faster.
But what about self-actualisation, the highest human aspiration in Maslow's hierarchy of needs? Here, AI has been less successful. Creative endeavours, such as writing a novel or making a scientific breakthrough, are seemingly more resistant to automation.
Arguably the crowning achievement of human creativity is mathematical insight. It catalysed the scientific revolution and underpins nearly all modern technologies. Yet, paradoxically, AI has the potential to impact mathematical insight even more than other creative domains. As we describe in a Nature World View article, there are two reasons for this.
First, there are no coincidences in mathematics. This is in stark contrast with the physical world, where all measurements have error, and false negatives and false positives abound. In mathematics, there are no false negatives—a single counterexample leaves a conjecture dead in the water. Likewise, there are no false positives. For example, Euler’s solution to the Basel problem states that the infinite sum 1/12 + 1/22 + 1/32 + … is equal to π2/6. This unexpected occurrence of π is no fluke. It points to a deep connection between number theory and the circumference of the circle.
The second reason AI will impact mathematical insight is that mathematical data is cheap. Primes, polyhedra, knots, groups and tessellations are abundant or easy to generate. This is in stark contrast with data in other fields in which AI can be transformative, such as medical diagnostics, drug discovery and material science, which requires costly experimental verification of the training data. For example, training AlphaFold to predict protein shapes required countless actual protein structures painstakingly deduced via crystallography.
Simply put, mathematical data is cheap, any pattern within it is meaningful, AI is the archetypal tool for pattern spotting, and mathematics is the universal language of patterns.
Our solution
To address this technological gap in the market, scientists at the London Institute for Mathematical Sciences created Irix, a company that combines exact mathematical data, machine learning, complex optimisation and metamathematics to create a platform for general mathematical insight.
We are pursuing two related products. The first is a relational database of mathematical patterns that teaches itself to find new relations. When the FBI introduced the Integrated Automated Fingerprint Identification System in 1999, the ability of law enforcement agencies to solve crimes improved dramatically. In a similar way, our database of mathematical patterns helps users identify mathematical structure in their own problems. This is important, because the same mathematical patterns show up in seemingly very different problems. When this happens, true statements applicable in one context carry over to the other context.
But our database goes further. It automatically finds new relations between different mathematical patterns by using transformer models to determine likely sequences of transformations between patterns, such as summing or convolving. By iterating this process, the relational database teaches itself.
The second product is a tool that machine-learns relations between attributes of families of mathematical objects. Examples of families include graphs, knots, groups, elliptic curves and cellular automata. Whereas for a human family, the attributes might be age, height and profession, for a family of knots, say, the attributes might be crossing number, total curvature and Jones polynomial.
We apply machine learning and complex optimisation techniques to find relations between these attributes of a family. In mathematics, where there are no coincidences, these relations are inherently meaningful. This contrasts with scientific data, where noise and measurement error often produce spurious relations. When one of our scientists applied this approach to elliptic curves—simple polynomial equations used in modern cryptography—he discovered deep new structural insights, which were reported in Quanta Magazine.
The two products can work in tandem. This is because attribute data for a given family of mathematical objects can be uploaded as a pattern to the relational database. Each time this happens, the probability of the relational database finding a new relation increases. This in turn can lead to new understanding of the original family of objects.
A VAST MARKET
The platform addresses a vast unmet need. While AI has transformed industries like defence and medicine, no system yet exists for general mathematical insight. The stakes are high: from computing to finance to synthetic biology, mathematical insights underpin most modern innovations. Irix will help companies in biotech, engineering, manufacturing and beyond to achieve better mathematical insight and find relations to known solutions.
Unlike language, law and even medicine, mathematical reasoning is universal across cultures and countries. This makes a platform for mathematical insight particularly resilient to consumer behaviours and changing external circumstances.
The market is also large—in excess of $⅓ trillion. While it’s difficult to quantify the cost of mathematical reasoning in industry, we can estimate it in research institutions. The cost of a typical theory paper is $300,000. Arxiv, the standard repository for papers in the mathematical sciences, receives 250,000 submissions annually. Assuming half of all papers are on Arxiv, the total cost of academic mathematical reasoning is $150 billion per year. For relevant businesses—more numerous than institutions—the cost is likely higher, not to mention their greater losses from misleading insights.
How we obtain mathematical insight has hardly changed since Newton invented calculus and the laws of motion 350 years ago. No other domain of human endeavour has been more resistant to technological improvement.
For centuries, mathematical insight has relied on paper and pencil to work out examples and manipulate formulae. Only two technological advances have had an impact. First, computers made it easier to work out examples and collect data. Second, symbolic computation software, such as Mathematica, expedited manipulating and solving equations. Together, these advances sped up mathematical discovery by perhaps 50%—a modest gain compared to the giant leaps in efficiency technology has delivered elsewhere. Achieving a step change in mathematical insight requires a radically new approach.
AN OPEN PROBLEM
When it comes to reducing toil, improving health and boosting security, artificial intelligence has been transformative. From text translation to medical imaging to autonomous flight, AI helps us solve problems better and faster.
But what about self-actualisation, the highest human aspiration in Maslow's hierarchy of needs? Here, AI has been less successful. Creative endeavours, such as writing a novel or making a scientific breakthrough, are seemingly more resistant to automation.
Arguably the crowning achievement of human creativity is mathematical insight. It catalysed the scientific revolution and underpins nearly all modern technologies. Yet, paradoxically, AI has the potential to impact mathematical insight even more than other creative domains. As we describe in a Nature World View article, there are two reasons for this.
First, there are no coincidences in mathematics. This is in stark contrast with the physical world, where all measurements have error, and false negatives and false positives abound. In mathematics, there are no false negatives—a single counterexample leaves a conjecture dead in the water. Likewise, there are no false positives. For example, Euler’s solution to the Basel problem states that the infinite sum 1/12 + 1/22 + 1/32 + … is equal to π2/6. This unexpected occurrence of π is no fluke. It points to a deep connection between number theory and the circumference of the circle.
The second reason AI will impact mathematical insight is that mathematical data is cheap. Primes, polyhedra, knots, groups and tessellations are abundant or easy to generate. This is in stark contrast with data in other fields in which AI can be transformative, such as medical diagnostics, drug discovery and material science, which requires costly experimental verification of the training data. For example, training AlphaFold to predict protein shapes required countless actual protein structures painstakingly deduced via crystallography.
Simply put, mathematical data is cheap, any pattern within it is meaningful, AI is the archetypal tool for pattern spotting, and mathematics is the universal language of patterns.
Our solution
To address this technological gap in the market, scientists at the London Institute for Mathematical Sciences created Irix, a company that combines exact mathematical data, machine learning, complex optimisation and metamathematics to create a platform for general mathematical insight.
We are pursuing two related products. The first is a relational database of mathematical patterns that teaches itself to find new relations. When the FBI introduced the Integrated Automated Fingerprint Identification System in 1999, the ability of law enforcement agencies to solve crimes improved dramatically. In a similar way, our database of mathematical patterns helps users identify mathematical structure in their own problems. This is important, because the same mathematical patterns show up in seemingly very different problems. When this happens, true statements applicable in one context carry over to the other context.
But our database goes further. It automatically finds new relations between different mathematical patterns by using transformer models to determine likely sequences of transformations between patterns, such as summing or convolving. By iterating this process, the relational database teaches itself.
The second product is a tool that machine-learns relations between attributes of families of mathematical objects. Examples of families include graphs, knots, groups, elliptic curves and cellular automata. Whereas for a human family, the attributes might be age, height and profession, for a family of knots, say, the attributes might be crossing number, total curvature and Jones polynomial.
We apply machine learning and complex optimisation techniques to find relations between these attributes of a family. In mathematics, where there are no coincidences, these relations are inherently meaningful. This contrasts with scientific data, where noise and measurement error often produce spurious relations. When one of our scientists applied this approach to elliptic curves—simple polynomial equations used in modern cryptography—he discovered deep new structural insights, which were reported in Quanta Magazine.
The two products can work in tandem. This is because attribute data for a given family of mathematical objects can be uploaded as a pattern to the relational database. Each time this happens, the probability of the relational database finding a new relation increases. This in turn can lead to new understanding of the original family of objects.
A VAST MARKET
The platform addresses a vast unmet need. While AI has transformed industries like defence and medicine, no system yet exists for general mathematical insight. The stakes are high: from computing to finance to synthetic biology, mathematical insights underpin most modern innovations. Irix will help companies in biotech, engineering, manufacturing and beyond to achieve better mathematical insight and find relations to known solutions.
Unlike language, law and even medicine, mathematical reasoning is universal across cultures and countries. This makes a platform for mathematical insight particularly resilient to consumer behaviours and changing external circumstances.
The market is also large—in excess of $⅓ trillion. While it’s difficult to quantify the cost of mathematical reasoning in industry, we can estimate it in research institutions. The cost of a typical theory paper is $300,000. Arxiv, the standard repository for papers in the mathematical sciences, receives 250,000 submissions annually. Assuming half of all papers are on Arxiv, the total cost of academic mathematical reasoning is $150 billion per year. For relevant businesses—more numerous than institutions—the cost is likely higher, not to mention their greater losses from misleading insights.