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Category: mathematics – Page 4

For much of history, many mathematicians—

Following thinkers like Aristotle—viewed infinity as a never-ending process rather than a completed object. In the late 19th century, Georg Cantor revolutionized this view by treating infinite sets as mathematical objects that could be compared and studied. His work showed that not all infinities are equal, and that there are infinitely many different sizes of infinity. While his ideas are foundational in modern mathematics, some philosophical schools, such as finitism and ultrafinitism, continue to question whether infinite objects meaningfully exist.

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    • TIMESTAMPS:

    • 00:00 — Potential vs. Actual Infinity
    • 03:12 — Cardinality and Aleph-Null
    • 06:12 — Diagonalization and Uncountability
    • 09:21 — ZFC and Logical Independence
    • 12:23 — Finitism and Ultrafinitism
    • 15:26 — Continuum Hypothesis Paradoxes
    • 16:00 — Foundational Mathematical Crisis

    LINKS MENTIONED:

    In Search of Ultimate-L [paper]: https://www.jstor.org/stable/44164514

    • Emily Riehl [TOE]: • Emily Riehl Makes Infinity Categories Elem…
    • Sir Roger Penrose [TOE]: • Roger Penrose: Quantum Theory Is Wrong, No…
    • Why Write? [article]: https://curtjaimungal.substack.com/p/.…

    ASSETS USED:

    • Infinity display: • 1h Infinity Symbol Animation for Backgroun…
    • Number counter: • Count up number 1–4000 overlay for edits

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    TIMESTAMPS:
    00:00 — Potential vs. Actual Infinity.
    03:12 — Cardinality and Aleph-Null.
    06:12 — Diagonalization and Uncountability.
    09:21 — ZFC and Logical Independence.
    12:23 — Finitism and Ultrafinitism.
    15:26 — Continuum Hypothesis Paradoxes.
    16:00 — Foundational Mathematical Crisis.

    LINKS MENTIONED:
    The Most Abused Theorem in Math [TOE]: • The Most Abused Theorem in Math (Gödel’s I…
    Dror Bar Natan [TOE]: • Dror Bar Natan: Knot Theory & Quantum Fiel…
    Hilbert’s Problems: https://mathworld.wolfram.com/Hilbert…
    The Independence of the Continuum Hypothesis [paper]: https://www.pnas.org/doi/pdf/10.1073/.…
    Peano arithmetic: https://ncatlab.org/nlab/show/Peano+a…
    Cantor’s Diagonal Argument: https://www.researchgate.net/publicat…
    Hartog’s Construction: paultaylor.eu/trans/HartogsF-wellord.pdf.
    Cohen’s Forcing Method: https://timothychow.net/forcing.pdf.
    Norman Wildberger [TOE]: • Norman Wildberger: The Problem with Infini…
    Woodin’s lecture: • The Continuum Hypothesis and the search fo…
    In Search of Ultimate-L [paper]: https://www.jstor.org/stable/44164514
    Emily Riehl [TOE]: • Emily Riehl Makes Infinity Categories Elem…
    Sir Roger Penrose [TOE]: • Roger Penrose: Quantum Theory Is Wrong, No…
    Why Write? [article]: https://curtjaimungal.substack.com/p/.…

    ASSETS USED:

    Continue reading “For much of history, many mathematicians—” | >

    Spatiotemporal light pulses could secure optical communication by masking data

    Researchers at Ben-Gurion University of the Negev have developed a new approach to secure optical communication that hides information in the physical structure of light, making it difficult for unauthorized parties to intercept or decode. The study addresses a growing challenge: advances in quantum computing are expected to weaken many of today’s encryption methods. While most security solutions rely on complex mathematical algorithms, this research adds protection earlier in the process—during the transmission of the signal itself.

    The research was led by Dr. Judith Kupferman and Prof. Shlomi Arnon from the School of Electrical and Computer Engineering at Ben-Gurion University of the Negev. The findings were published in Optical and Quantum Electronics.

    The researchers propose a communication method based on specially shaped light pulses, known as spatiotemporal optical vortices. These light beams are designed so that their key features are not visible in standard measurements.

    Self-propulsion or slow diffusion: How bacteria, cells, and colloids respond to stimuli

    What physical processes govern the movement of microscopic structures capable of interacting with their environment? The answer lies in two mechanisms: self-propulsion, to escape unfavorable locations; and slow diffusion, to move toward more advantageous ones. This is the finding of scientists Jacopo Romano and Andrea Gambassi from SISSA-Scuola Internazionale Superiore di Studi Avanzati in their new study published in Physical Review Letters.

    In their work, the researchers combined computer simulations with mathematical calculations, taking inspiration from nature. It is well known that feedback-driven motion underlies the behavior of various microorganisms, which analyze incoming and outgoing signals and adapt their direction of movement accordingly. The study reproduces the physical behavior of natural and synthetic agents in two distinct scenarios: when a specific destination must be avoided based on signals, and when it must instead be reached.

    The researchers found that in the first case, a process of “superdiffusion” occurs, with accelerated motion, while in the second case a subdiffusive process takes place, with much slower movement. These findings provide important insights for the design of smart particles capable of moving at the microscale, with potential applications in medicine, particularly for more efficient drug delivery.

    An example of how AI struggles to solve a simple ARC-AGI Benchmark challenge question

    For Context: OpenAI has recently introduced two new AI models, o3 and o3-mini, designed to enhance reasoning capabilities in complex tasks such as advanced mathematics, science, and coding. These models represent a significant advancement over their predecessor, o1, which was released in September 2024.

    Wired.

    Physicists Just Linked This 160 Year-Old Math Problem To Black Holes

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    The Riemann Hypothesis is an open problem in maths which – if proved correct – would show us a pattern in prime numbers. The zeta function, a central part of the hypothesis, has been linked to quantum mechanics, and recently a group of physicists linked it to gravitational equations associated with black holes. What does this mean, exactly? Let’s take a look.

    Paper: https://link.springer.com/article/10… mugs, posters and more: ➜ https://sabines-store.dashery.com/ 💌 Support me on Donorbox ➜ https://donorbox.org/swtg 👉 Transcript with links to references on Patreon ➜ / sabine 📝 Transcripts and written news on Substack ➜ https://sciencewtg.substack.com/ 📩 Free weekly science newsletter ➜ https://sabinehossenfelder.com/newsle… 👂 Audio only podcast ➜ https://open.spotify.com/show/0MkNfXl… 🔗 Join this channel to get access to perks ➜ / @sabinehossenfelder 📚 Buy my book ➜ https://amzn.to/3HSAWJW #science #sciencenews #physics #maths The Riemann hypothesis is a significant open problem in mathematics, deeply intertwined with number theory and its implications for physics. This video explores how the riemann zeta function, a central element of the hypothesis, connects to fundamental concepts like black hole physics and quantum gravity. Discover the ongoing mathematical research that seeks to solve this enduring mystery…

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    #science #sciencenews #physics #maths.

    The Riemann hypothesis is a significant open problem in mathematics, deeply intertwined with number theory and its implications for physics. This video explores how the riemann zeta function, a central element of the hypothesis, connects to fundamental concepts like black hole physics and quantum gravity. Discover the ongoing mathematical research that seeks to solve this enduring mystery.

    Continue reading “Physicists Just Linked This 160 Year-Old Math Problem To Black Holes” | >

    Math model reveals how life may have switched on from Earth’s primordial soup

    Isolating the first spark of life on Earth is a matter of biology, geology, and chemistry—but it’s also an amazing math problem. At least, that’s how Varun Varanasi viewed it when he was a Yale undergraduate. The question, in a nutshell, is this: How did the primordial soup of interacting molecules on the Earth’s surface billions of years ago transform itself from complete chaos to an organized system of self-sustaining, reproducing chemicals? Did this occur gradually over millions of years, or was it abrupt?

    Rapid method uncovers hidden structures in materials—including elusive quasicrystals

    An international team of scientists, including researchers from Loughborough University, has developed a method to dramatically speed up the discovery and design of advanced materials. The study, published in Physical Review Letters, shows how the new approach can map complex phase diagrams in as little as a day—rather than weeks or months—and pinpoint where important structures, including crystals and quasicrystals, are likely to form.

    The method will enable scientists to “scout ahead” and identify where promising structures are likely to form and the conditions needed to create them, rather than using a trial-and-error approach. It could help accelerate the development of advanced materials and technologies that harness the unique properties of quasicrystal structures.

    “Our approach is a day’s work for an expert—it’s much faster,” said Professor Andrew Archer, an expert in applied mathematics and theoretical physics at Loughborough University and one of the paper’s authors.

    Math Professor Wrote Wrong Equation on the Board to Test a Black Student—But He Was a Genius Student

    What if creativity wasn’t magic—but math?
    In this video, we explore the mathematics of creativity through psychology, philosophy, and science. From Dean Keith Simonton’s law of large numbers, Margaret Boden’s theory of combinational creativity, Zipf’s Law, Malcolm Gladwell’s 10,000-hour curve, and even cellular automata—we break down how imagination follows hidden equations.

    Whether you’re a student, teacher, scientist, engineer, or philosopher, this video will change how you think about art, science, and human innovation.

    Chapters:
    00:00 – Intro: Is Creativity Random?
    00:34 – The Law of Large Numbers
    01:42 – Zipf’s Law of Ideas
    02:33 – Combinational Creativity (Boden)
    03:15 – Time & Growth (Gladwell)
    03:58 – Edge of Chaos (Complexity Theory)
    04:48 – The Formula for Creativity.

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    Sofia Kovalevskaya: The Girl Who Wouldn’t Give Up on Math

    “It is impossible to be a mathematician without being a poet in soul.”-Sofia Kovalevskaya

    We don’t often think of math as something that’s “dangerous” or “forbidden”; after all, what could be so dangerous about numbers? Russian-born Sofia Kovalevskaya was told at numerous points during her life that she had to stop studying math, that girls weren’t good enough, they weren’t allowed to go to school, or teach classes, edit magazines or win awards. Sofia Kovalevskaya never gave in to the couldn’t’s or wouldn’t’s. She fought time and again for her right to continue learning and teaching, eventually becoming one of the most celebrated mathematicians of her century and the first woman professor of a northern European University. Today, we celebrate Sofia and all the young mathematicians who overcome great odds!

    When Sofia Kovalevskaya was a little girl in the early 1850’s, her room wasn’t wallpapered with flowers or meadowscapes, it was covered in pages and pages of math lecture not es. She would stare at the pages filled with differential and integral analysis, and while she didn’t understand exactly what she saw, Sofia saw beauty in the calculations.

    Universal surface-growth law confirmed in two dimensions after 40 years

    Crystals, bacterial colonies, flame fronts: the growth of surfaces was first described in the 1980s by the Kardar–Parisi–Zhang equation. Since then, it has been regarded as a fundamental model in physics, with implications for mathematics, biology, and computer science.

    Now—40 years later—a Würzburg-based research team from the Cluster of Excellence ctd.qmat has achieved the first experimental demonstration of KPZ behavior on 2D surfaces in space and time.

    This was made possible by sophisticated materials engineering and a bold experimental approach: researchers injected polaritons—hybrid particles composed of light and matter—into the material. The results have been published in Science.

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