Sir Roger Penrose, a name synonymous with genius, has tirelessly pursued the secrets of the universe with the fervour of a true renaissance seer. His intellectual contributions span a breathtaking range, from the intricate beauty of Penrose tilings to the vast expanse of cosmology, and even the enigmatic depths of human consciousness.
A non-technical introduction to platonism in the philosophy of mathematics.
Philosophy of mathematics is important, especially for philosophers interested in metaphysics. Suppose, for instance, you have nominalist tendencies, and you argue against the existence of abstract objects. Well, probably the most important kind of abstract objects are found in mathematics. Any serious nominalist needs to give an account of them.
Yet philosophy of mathematics is also, for obvious reasons, quite technical, and it can be pretty daunting for those who have less mathematical training. Nevertheless, I think the basic arguments can be made accessible to anyone who’s interested, and that’s what I’ve tried to do in this video.
For further reading on phil of mathematics, I recommend: “Thinking About Mathematics” by Stewart Shapiro and “Introducing Philosophy of Mathematics” by Michele Friend. These are both fairly orthodox introductions. For an introduction that focuses more on contemporary issues (it has just a few pages devoted to formalism, logicism, and intuitionism, yet a whole chapter on paraconsistent mathematics), I recommend “An Introduction to the Philosophy of Mathematics” by Mark Colyvan.
There is some debate about how exactly to formulate the Quine-Putnam Indispensability Argument. In this video, I’ve followed Colyvan (see his aforementioned “Introduction”).
Re 43:26: I’m not sure why I said that. Obviously, Von Neumann did not hold the absurd belief that every ordinal *is* the ordinal before it; rather he believed that every ordinal is *the set of* all ordinals before it.
Users of Google’s Chrome browser can rest easy knowing that their surfing is secure, thanks in part to cryptographer Joppe Bos. He’s coauthor of a quantum-secure encryption algorithm that was adopted as a standard by the U.S. National Institute of Standards and Technology (NIST) in August and is already being implemented in a wide range of technology products, including Chrome.
Rapid advances in quantum computing have stoked fears that future devices may be able to break the encryption used by most modern technology. These approaches to encryption typically rely on mathematical puzzles that are too complex for classical computers to crack. But quantum computers can exploit quantum phenomena like superposition and entanglement to compute these problems much faster, and a powerful enough machine should be able to break current encryption.
Simulations deliver hints on how the multiverse produced according to the many-worlds interpretation of quantum mechanics might be compatible with our stable, classical Universe.
We understand quantum mechanics well enough to make stunningly accurate predictions, ranging from atomic spectra to the structure of neutron stars, and to successfully exploit these predictions in devices such as lasers, MRI machines, and tunneling microscopes. Yet there is no generally accepted explanation of how the solid reality of such devices—or of objects such as cats, moons, and people—arise from a nebulous quantum wave in an abstract mathematical space. Some physicists prefer to ignore the problem, suggesting that we should just “shut up and calculate!” Others seek answers by modifying quantum theory in various ways or by searching for ways to explain how stable structures can emerge from quantum theory itself.
A new artificial intelligence (AI) model has just achieved human-level results on a test designed to measure “general intelligence.”
On December 20, OpenAI’s o3 system scored 85% on the ARC-AGI benchmark, well above the previous AI best score of 55% and on par with the average human score. It also scored well on a very difficult mathematics test.
Creating artificial general intelligence, or AGI, is the stated goal of all the major AI research labs. At first glance, OpenAI appears to have at least made a significant step towards this goal.
Here’s one definition of science: it’s essentially an iterative process of building models with ever-greater explanatory power.
A model is just an approximation or simplification of how we think the world works. In the past, these models could be very simple, as simple in fact as a mathematical formula. But over time, they have evolved and scientists have built increasingly sophisticated simulations of the world as new data has become available.
A computer model of the Earth’s climate can show us temperatures will rise as we continue to release greenhouse gases into the atmosphere. Models can also predict how infectious disease will spread in a population, for example.
The manic pace of sharing, storing, securing, and serving data has a manic price—power consumption. To counter this, Virginia Tech mathematicians are leveraging algebraic geometry to target the inefficiencies of data centers.
“We as individuals generate tons of data all the time, not to mention what large companies are producing,” said Gretchen Matthews, mathematics professor and director of the Southwest Virginia node of the Commonwealth Cyber Initiative. “Backing up that data can mean replicating and storing twice or three times as much information if we don’t consider smart alternatives.”
Instead of energy-intensive data replication, Matthews and Hiram Lopez, assistant professor of mathematics, explored using certain algebraic structures to break the information into pieces and spread it out among servers in close proximity to each other. When one server goes down, the algorithm can poll the neighboring servers until it recovers the missing data.
Skoltech researchers have proposed novel mathematical equations that describe the behavior of aggregating particles in fluids. This bears on natural and engineering processes as diverse as rain and snow formation, the emergence of planetary rings, and the flow of fluids and powders in pipes.
Reported in Physical Review Letters, the new equations eliminate the need for juggling two sets of equations that had to be used in conjunction, which led to unacceptable errors for some applications.
Fluid aggregation is involved in many processes. In the atmosphere, water droplets agglomerate into rain, and ice microcrystals into snow. In space, particles orbiting giant planets come together to form rings like those of Saturn.
Macquarie University researchers have demonstrated how ordinary supermarket grapes can enhance the performance of quantum sensors, potentially leading to more efficient quantum technologies.
The study, published in Physical Review Applied on 20 December 2024, shows that pairs of grapes can create strong localized magnetic field hotspots of microwaves which are used in quantum sensing applications—a finding that could help develop more compact and cost-effective quantum devices.
“While previous studies looked at the electrical fields causing the plasma effect, we showed that grape pairs can also enhance magnetic fields, which are crucial for quantum sensing applications,” says lead author Ali Fawaz, a quantum physics Ph.D. candidate at Macquarie University.
