<![CDATA[MKLab - EXPOSITION]]> https://mklab.gr/ Wed, 28 Oct 2026 00:30:03 +0000 MyBB <![CDATA[Pascal's wager]]> https://mklab.gr/showthread.php?tid=2213 Sun, 27 Sep 2026 21:20:01 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=2213 Pensées. The argument frames belief in God as a life-defining gamble, arguing that a rational person should adopt a lifestyle consistent with God's existence and try to believe: if God does not exist, the believer only loses some finite pleasures, but if God does exist, the believer gains immeasurably while avoiding infinite loss. It is generally regarded as the earliest formal use of decision theory, and it also anticipates ideas later associated with existentialism, pragmatism, and voluntarism.

The core of Pascal's reasoning is that reason alone cannot settle whether God exists, so the question is effectively a coin toss — yet everyone is "already embarked" and must live as though one side or the other is true, since abstaining isn't an option. Given that framing, Pascal compares the stakes: a finite cost (foregone earthly pleasures) against an infinite potential reward (eternal happiness) or infinite potential loss (eternal damnation). Because any finite stake is trivial next to an infinite expected gain, he concludes it is rational to wager that God exists. 

This can be formalized as a decision matrix where belief carries a possible infinite payoff and disbelief a possible infinite loss — and as long as there's any positive probability God exists, betting on belief has the higher expected value. Pascal also acknowledged that genuine faith can't simply be willed into being on the strength of this calculation, so he recommended a practical path: adopting the outward habits of religious practice (prayer, ritual) in hopes that behaving as a believer would gradually produce real belief.

The wager has drawn extensive criticism since Pascal's time. Voltaire dismissed using self-interest as a "proof" of God as intellectually indecent. The "argument from inconsistent revelations" (raised in different forms by Diderot and later J. L. Mackie) points out that many religions each claim exclusive routes to salvation, so betting on any one particular god doesn't obviously carry positive expected value — a problem Pascal anticipated but treated as a rhetorical trap he refused to engage with in depth, focusing his rebuttal on Christianity's supposed distinctiveness from paganism and Islam.

 Others, including Étienne Souriau, noted the wager assumes God would honor the "bet" at all. Modern critics like Richard Dawkins argue that manufacturing belief purely for personal gain is both dishonest and something an omniscient God would presumably see through, while William James made a similar point about the psychological implausibility of "willing" oneself into faith via calculation. Theologically, critics also note that mere intellectual assent isn't considered sufficient for salvation in either Catholic or Protestant doctrine — genuine "faith" requires more than a wagered belief.

Beyond its original theological context, the wager's basic logic — small, certain cost versus a small-probability but enormous payoff or penalty — has echoed widely. Similar arguments appear earlier in figures like the Stoic Marcus Aurelius, early Buddhist texts, and Islamic theologians such as al-Juwayni, and later in secular offshoots such as the "Atheist's Wager," Nick Bostrom's "Pascal's Mugging," and the "Roko's Basilisk" thought experiment. 

The structure has also been explicitly applied to modern policy debates: since the early 1990s, scholars and figures like Warren Buffett have compared the wager's logic to climate change, arguing that even a small probability of catastrophic, hard-to-reverse consequences can justify acting now — though critics note climate change, unlike Pascal's God, has actual scientific evidence behind it and its potential costs, while severe, are not literally infinite.

Key takeaways:
  • Pascal's wager argues that believing in God is the rational bet: finite potential cost versus infinite potential reward (or avoided infinite loss).
  • It's considered the first formal application of decision theory to a question of belief, and predates related ideas in existentialism and pragmatism.
  • Pascal argued people can't opt out of the wager — living itself forces an implicit choice about how to act regarding God's existence.
  • Since reason can't produce belief on command, Pascal suggested cultivating faith through religious practice and habit rather than pure logical persuasion.
  • Major objections include the "many religions" problem (which god to bet on?), the charge of promoting insincere/manufactured belief, and doubts that God would accept or reward a self-interested bet.
  • The same cost-benefit logic has been echoed across cultures and eras (Stoicism, Buddhism, Islamic theology) and adapted into secular arguments, including modern comparisons to climate-change risk decisions.

ARTICLE]]> Pensées. The argument frames belief in God as a life-defining gamble, arguing that a rational person should adopt a lifestyle consistent with God's existence and try to believe: if God does not exist, the believer only loses some finite pleasures, but if God does exist, the believer gains immeasurably while avoiding infinite loss. It is generally regarded as the earliest formal use of decision theory, and it also anticipates ideas later associated with existentialism, pragmatism, and voluntarism.

The core of Pascal's reasoning is that reason alone cannot settle whether God exists, so the question is effectively a coin toss — yet everyone is "already embarked" and must live as though one side or the other is true, since abstaining isn't an option. Given that framing, Pascal compares the stakes: a finite cost (foregone earthly pleasures) against an infinite potential reward (eternal happiness) or infinite potential loss (eternal damnation). Because any finite stake is trivial next to an infinite expected gain, he concludes it is rational to wager that God exists. 

This can be formalized as a decision matrix where belief carries a possible infinite payoff and disbelief a possible infinite loss — and as long as there's any positive probability God exists, betting on belief has the higher expected value. Pascal also acknowledged that genuine faith can't simply be willed into being on the strength of this calculation, so he recommended a practical path: adopting the outward habits of religious practice (prayer, ritual) in hopes that behaving as a believer would gradually produce real belief.

The wager has drawn extensive criticism since Pascal's time. Voltaire dismissed using self-interest as a "proof" of God as intellectually indecent. The "argument from inconsistent revelations" (raised in different forms by Diderot and later J. L. Mackie) points out that many religions each claim exclusive routes to salvation, so betting on any one particular god doesn't obviously carry positive expected value — a problem Pascal anticipated but treated as a rhetorical trap he refused to engage with in depth, focusing his rebuttal on Christianity's supposed distinctiveness from paganism and Islam.

 Others, including Étienne Souriau, noted the wager assumes God would honor the "bet" at all. Modern critics like Richard Dawkins argue that manufacturing belief purely for personal gain is both dishonest and something an omniscient God would presumably see through, while William James made a similar point about the psychological implausibility of "willing" oneself into faith via calculation. Theologically, critics also note that mere intellectual assent isn't considered sufficient for salvation in either Catholic or Protestant doctrine — genuine "faith" requires more than a wagered belief.

Beyond its original theological context, the wager's basic logic — small, certain cost versus a small-probability but enormous payoff or penalty — has echoed widely. Similar arguments appear earlier in figures like the Stoic Marcus Aurelius, early Buddhist texts, and Islamic theologians such as al-Juwayni, and later in secular offshoots such as the "Atheist's Wager," Nick Bostrom's "Pascal's Mugging," and the "Roko's Basilisk" thought experiment. 

The structure has also been explicitly applied to modern policy debates: since the early 1990s, scholars and figures like Warren Buffett have compared the wager's logic to climate change, arguing that even a small probability of catastrophic, hard-to-reverse consequences can justify acting now — though critics note climate change, unlike Pascal's God, has actual scientific evidence behind it and its potential costs, while severe, are not literally infinite.

Key takeaways:
  • Pascal's wager argues that believing in God is the rational bet: finite potential cost versus infinite potential reward (or avoided infinite loss).
  • It's considered the first formal application of decision theory to a question of belief, and predates related ideas in existentialism and pragmatism.
  • Pascal argued people can't opt out of the wager — living itself forces an implicit choice about how to act regarding God's existence.
  • Since reason can't produce belief on command, Pascal suggested cultivating faith through religious practice and habit rather than pure logical persuasion.
  • Major objections include the "many religions" problem (which god to bet on?), the charge of promoting insincere/manufactured belief, and doubts that God would accept or reward a self-interested bet.
  • The same cost-benefit logic has been echoed across cultures and eras (Stoicism, Buddhism, Islamic theology) and adapted into secular arguments, including modern comparisons to climate-change risk decisions.

ARTICLE]]> <![CDATA[Careers in the MATHEMATICAL SCIENCES [SIAM]]]> https://mklab.gr/showthread.php?tid=2085 Wed, 23 Sep 2026 19:29:10 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=2085 2022 SIAM Careers in the Mathematical Sciences brochure explains how mathematics, computational science, and data science translate into careers far beyond traditional academic mathematics. It emphasizes that jobs using mathematical expertise often do not carry the title mathematician and can involve specialties such as data science, machine learning, quantitative analysis, software engineering, operations research, finance, healthcare, climate science, manufacturing, and transportation. 

The brochure highlights real-world problems where mathematical modeling and computation are central—for example, optimizing airline schedules, designing clinical trials, analyzing genomes, managing financial risk, improving semiconductor manufacturing, and personalizing medical treatment.

A major message is that students should combine a strong mathematical foundation—including differential equations, probability, combinatorics, linear algebra, abstraction, and programming—with the ability to apply those skills to practical problems and communicate results effectively. SIAM recommends internships, work-study, research experiences, career fairs, and openness to job titles that may not explicitly mention mathematics. 

The brochure also profiles more than 20 professionals working in industry, government, research laboratories, technology, healthcare, finance, and other sectors, illustrating the breadth of career paths available to people trained in the mathematical sciences. 

ARTICLE [PDF]]]> 2022 SIAM Careers in the Mathematical Sciences brochure explains how mathematics, computational science, and data science translate into careers far beyond traditional academic mathematics. It emphasizes that jobs using mathematical expertise often do not carry the title mathematician and can involve specialties such as data science, machine learning, quantitative analysis, software engineering, operations research, finance, healthcare, climate science, manufacturing, and transportation. 

The brochure highlights real-world problems where mathematical modeling and computation are central—for example, optimizing airline schedules, designing clinical trials, analyzing genomes, managing financial risk, improving semiconductor manufacturing, and personalizing medical treatment.

A major message is that students should combine a strong mathematical foundation—including differential equations, probability, combinatorics, linear algebra, abstraction, and programming—with the ability to apply those skills to practical problems and communicate results effectively. SIAM recommends internships, work-study, research experiences, career fairs, and openness to job titles that may not explicitly mention mathematics. 

The brochure also profiles more than 20 professionals working in industry, government, research laboratories, technology, healthcare, finance, and other sectors, illustrating the breadth of career paths available to people trained in the mathematical sciences. 

ARTICLE [PDF]]]> <![CDATA[A Discussion with Freeman Dyson]]> https://mklab.gr/showthread.php?tid=1994 Tue, 15 Sep 2026 19:45:16 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1994 Freeman Dyson, conducted by Michael Th. Rassias and published in the EMS Newsletter shortly before Dyson’s death in February 2020. Dyson reflects on his unusual path from pure mathematics to theoretical physics: as a young mathematician he worked with H. Davenport and proved Minkowski’s conjecture in four dimensions, but later became fascinated by physics after encountering Nicholas Kemmer at Cambridge. 

He describes the profound influence of mathematicians such as A. S. Besicovitch and G. H. Hardy, as well as physicists including Wolfgang Pauli and Richard Feynman. Dyson also explains his opposition to the modern Ph.D. system, which he considered unnecessarily rigid and socially unequal, and emphasizes the value of following intellectual curiosity rather than conventional academic paths.

The second half of the interview explores Dyson’s broader philosophy of science and technology. He argues that quantum computers should be viewed as “quantum subroutines” requiring classical systems and human interaction, and warns that practical security will remain vulnerable to human error. He is critical of contemporary physics when it becomes dominated by enormous experimental projects and theories lacking testable predictions, while praising mathematics for its unpredictability: the next major breakthrough, he believes, will inevitably be a surprise.

 On AI, Dyson points to Norbert Wiener’s 1950 work as an exceptionally prescient analysis of the potential benefits and dangers of intelligent machines, arguing that society has still failed to develop adequate safeguards. Overall, the interview portrays Dyson as an intensely curious, interdisciplinary scientist who valued imagination, independent thinking, and the unexpected connections between mathematics, physics, biology, and other fields.


ARTICLE [PDF]]]> Freeman Dyson, conducted by Michael Th. Rassias and published in the EMS Newsletter shortly before Dyson’s death in February 2020. Dyson reflects on his unusual path from pure mathematics to theoretical physics: as a young mathematician he worked with H. Davenport and proved Minkowski’s conjecture in four dimensions, but later became fascinated by physics after encountering Nicholas Kemmer at Cambridge. 

He describes the profound influence of mathematicians such as A. S. Besicovitch and G. H. Hardy, as well as physicists including Wolfgang Pauli and Richard Feynman. Dyson also explains his opposition to the modern Ph.D. system, which he considered unnecessarily rigid and socially unequal, and emphasizes the value of following intellectual curiosity rather than conventional academic paths.

The second half of the interview explores Dyson’s broader philosophy of science and technology. He argues that quantum computers should be viewed as “quantum subroutines” requiring classical systems and human interaction, and warns that practical security will remain vulnerable to human error. He is critical of contemporary physics when it becomes dominated by enormous experimental projects and theories lacking testable predictions, while praising mathematics for its unpredictability: the next major breakthrough, he believes, will inevitably be a surprise.

 On AI, Dyson points to Norbert Wiener’s 1950 work as an exceptionally prescient analysis of the potential benefits and dangers of intelligent machines, arguing that society has still failed to develop adequate safeguards. Overall, the interview portrays Dyson as an intensely curious, interdisciplinary scientist who valued imagination, independent thinking, and the unexpected connections between mathematics, physics, biology, and other fields.


ARTICLE [PDF]]]> <![CDATA[Are old mathematicians useless?]]> https://mklab.gr/showthread.php?tid=1988 Tue, 15 Sep 2026 19:12:01 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1988 “Are old mathematicians useless?” challenges G. H. Hardy’s famous claim that mathematics is essentially a “young man’s game” and that major original advances rarely come after age 50. Pietsch argues that such an age limit is unjustified. He gives numerous counterexamples of mathematicians who produced important work well after 50, including Poincaré, Cartan, Schwartz, Calabi, Atiyah, Erdős, and others. Particularly striking is the range of achievements he lists: major results were produced at ages 58, 62, 69, 72, 78 and beyond, while Louis Vietoris published a paper at 103. The evidence therefore suggests that although mathematical productivity may peak relatively early for many researchers, exceptional creativity and substantial contributions can continue throughout old age. 

Pietsch then makes the argument personal, describing his own “second mathematical life.” He continued developing new mathematics decades after beginning his career, including work submitted at ages 82 and 89. His broader message is that mathematical research should not be viewed solely as a competition for youthful breakthroughs: curiosity, enjoyment, experience, and love of mathematics can remain valuable throughout life. 

At the same time, he acknowledges that some older mathematicians may lose enthusiasm or mental energy and should decide individually whether to continue. Even when active research becomes difficult, however, senior mathematicians have an important role in transmitting knowledge, ideas, open problems, and experience to younger generations. His final appeal is therefore not to impose an age-based notion of usefulness, but to respect and appreciate the contributions of mathematicians at every stage of their careers. 

ARTICLE [PDF]]]> “Are old mathematicians useless?” challenges G. H. Hardy’s famous claim that mathematics is essentially a “young man’s game” and that major original advances rarely come after age 50. Pietsch argues that such an age limit is unjustified. He gives numerous counterexamples of mathematicians who produced important work well after 50, including Poincaré, Cartan, Schwartz, Calabi, Atiyah, Erdős, and others. Particularly striking is the range of achievements he lists: major results were produced at ages 58, 62, 69, 72, 78 and beyond, while Louis Vietoris published a paper at 103. The evidence therefore suggests that although mathematical productivity may peak relatively early for many researchers, exceptional creativity and substantial contributions can continue throughout old age. 

Pietsch then makes the argument personal, describing his own “second mathematical life.” He continued developing new mathematics decades after beginning his career, including work submitted at ages 82 and 89. His broader message is that mathematical research should not be viewed solely as a competition for youthful breakthroughs: curiosity, enjoyment, experience, and love of mathematics can remain valuable throughout life. 

At the same time, he acknowledges that some older mathematicians may lose enthusiasm or mental energy and should decide individually whether to continue. Even when active research becomes difficult, however, senior mathematicians have an important role in transmitting knowledge, ideas, open problems, and experience to younger generations. His final appeal is therefore not to impose an age-based notion of usefulness, but to respect and appreciate the contributions of mathematicians at every stage of their careers. 

ARTICLE [PDF]]]> <![CDATA[“If, and only if” in mathematics]]> https://mklab.gr/showthread.php?tid=1986 Tue, 15 Sep 2026 19:06:29 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1986 “If, and only if” in mathematics by Omid Ali Shahni Karamzadeh argues that mathematicians should pay much more attention to the converse of mathematical statements. Many familiar theorems are traditionally written as “if $P$, then $Q$,” even when a stronger and more informative “if and only if” formulation is possible. 

The author calls such results “good complete” (gc-) theorems and argues that expressing both directions can deepen understanding, suggest proofs, reveal hidden connections, and sometimes turn difficult problems into almost immediate consequences. Examples include results from Euclidean geometry, Fermat’s Last Theorem, Morley’s theorem, and a characterization of the irrationality of $\sqrt{2}$. 

The article then develops this philosophy through examples from analysis, algebra, and topology. The Bolzano–Weierstrass theorem, for instance, is reformulated to characterize exactly which subsets of $\mathbb{R}^n$ have limit points. Similar stronger equivalences are presented for commutative rings and topological spaces. 

The author emphasizes that not every “if and only if” statement automatically qualifies as a genuine gc-theorem: one direction may be trivial, artificial, or lack a motivated proof. Ultimately, the central message is methodological: when encountering a mathematical result, actively investigate its converse. Doing so can produce stronger theorems, expose unexpected relationships, and sometimes lead to simpler or more illuminating proofs.


ARTICLE [PDF]]]> “If, and only if” in mathematics by Omid Ali Shahni Karamzadeh argues that mathematicians should pay much more attention to the converse of mathematical statements. Many familiar theorems are traditionally written as “if $P$, then $Q$,” even when a stronger and more informative “if and only if” formulation is possible. 

The author calls such results “good complete” (gc-) theorems and argues that expressing both directions can deepen understanding, suggest proofs, reveal hidden connections, and sometimes turn difficult problems into almost immediate consequences. Examples include results from Euclidean geometry, Fermat’s Last Theorem, Morley’s theorem, and a characterization of the irrationality of $\sqrt{2}$. 

The article then develops this philosophy through examples from analysis, algebra, and topology. The Bolzano–Weierstrass theorem, for instance, is reformulated to characterize exactly which subsets of $\mathbb{R}^n$ have limit points. Similar stronger equivalences are presented for commutative rings and topological spaces. 

The author emphasizes that not every “if and only if” statement automatically qualifies as a genuine gc-theorem: one direction may be trivial, artificial, or lack a motivated proof. Ultimately, the central message is methodological: when encountering a mathematical result, actively investigate its converse. Doing so can produce stronger theorems, expose unexpected relationships, and sometimes lead to simpler or more illuminating proofs.


ARTICLE [PDF]]]> <![CDATA[Ignoramus et ignorabimus]]> https://mklab.gr/showthread.php?tid=1976 Tue, 15 Sep 2026 18:00:26 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1976 Ignoramus et ignorabimus is a Latin phrase meaning “we do not know and we will not know.” It was popularized by German physiologist Emil du Bois-Reymond in his 1872 lecture The Limits of Science. He argued that science has fundamental boundaries: some questions may be permanently beyond human understanding, particularly the ultimate nature of matter and energy, the origin of motion, and the nature of conscious sensation. He later presented these issues as part of his famous “seven world riddles,” distinguishing problems that science could investigate from those he considered fundamentally unknowable. 

The claim provoked a major philosophical and scientific debate. David Hilbert rejected this pessimistic view, famously declaring that “in mathematics there is no ignorabimus” and expressing confidence that human reasoning could eventually solve mathematical problems. However, Gödel's incompleteness theorems in 1931 demonstrated that sufficiently powerful formal mathematical systems necessarily contain statements that cannot be proved or disproved within those systems, giving a new and more nuanced meaning to the idea of limits to knowledge. The debate therefore became a central question in the philosophy of science: are there intrinsic limits to what humanity can know, or are today's unknowns merely problems awaiting better methods and deeper understanding? 

ARTICLE]]> Ignoramus et ignorabimus is a Latin phrase meaning “we do not know and we will not know.” It was popularized by German physiologist Emil du Bois-Reymond in his 1872 lecture The Limits of Science. He argued that science has fundamental boundaries: some questions may be permanently beyond human understanding, particularly the ultimate nature of matter and energy, the origin of motion, and the nature of conscious sensation. He later presented these issues as part of his famous “seven world riddles,” distinguishing problems that science could investigate from those he considered fundamentally unknowable. 

The claim provoked a major philosophical and scientific debate. David Hilbert rejected this pessimistic view, famously declaring that “in mathematics there is no ignorabimus” and expressing confidence that human reasoning could eventually solve mathematical problems. However, Gödel's incompleteness theorems in 1931 demonstrated that sufficiently powerful formal mathematical systems necessarily contain statements that cannot be proved or disproved within those systems, giving a new and more nuanced meaning to the idea of limits to knowledge. The debate therefore became a central question in the philosophy of science: are there intrinsic limits to what humanity can know, or are today's unknowns merely problems awaiting better methods and deeper understanding? 

ARTICLE]]> <![CDATA[The Drake equation]]> https://mklab.gr/showthread.php?tid=1921 Wed, 09 Sep 2026 23:10:56 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1921
Jørgen Veisdal’s article explains the Drake equation, proposed by astronomer Frank Drake in 1961 as a framework for estimating how many technologically advanced extraterrestrial civilizations might currently be detectable in the Milky Way. The equation is
$N = R_* \times f_p \times n_e \times f_l \times f_i \times f_c \times L$,
where $R_*$ is the rate of star formation, $f_p$ the fraction of stars with planets, $n_e$ the number of potentially habitable planets per planetary system, $f_l$ the fraction on which life develops, $f_i$ the fraction that develops intelligence, $f_c$ the fraction producing detectable communications, and $L$ the duration for which such civilizations remain detectable. The equation was never intended to give a precise answer; rather, Drake devised it to organize the major scientific uncertainties involved in the search for extraterrestrial intelligence. Early optimistic assumptions produced estimates ranging from roughly 20 to tens of millions of communicating civilizations, illustrating how sensitive the result is to the chosen parameters.
 
Astronomical observations have since improved some of these estimates. We now know that planets appear to be extremely common, and studies using Kepler data have suggested that the Milky Way may contain billions of roughly Earth-sized planets in habitable zones. However, the biological and sociological factors—$f_l$, $f_i$, $f_c$, and especially $L$—remain largely speculative because Earth provides essentially our only example of life and technological civilization. Using more modern astronomical estimates while assigning intermediate values such as $0.5$ to the unknown factors, the article obtains an illustrative estimate of about 46 communicating extraterrestrial civilizations in the Milky Way. The author stresses that this number should not be interpreted as a scientific measurement; it mainly demonstrates how assumptions propagate through the Drake equation.
 
The article concludes by connecting the Drake equation with the Fermi paradox: if potentially habitable planets are abundant and technological civilizations are reasonably common, why have we seen no convincing evidence of them? Given the age and enormous number of stars in the Galaxy, even relatively slow interstellar expansion might allow a civilization to spread across the Milky Way in a few million years. Thus the tension between large Drake-equation estimates and the absence of observable extraterrestrials leads to Fermi’s famous question: “Where is everybody?”

Key takeaways:
  • The Drake equation is a framework for estimating the number of detectable extraterrestrial civilizations in the Milky Way.
  • It is not a precise prediction; its value depends heavily on uncertain assumptions.
  • Astronomy has improved estimates for factors such as the number of stars, planets, and potentially habitable worlds.
  • The biggest uncertainties remain biological and technological: how often life, intelligence, communication technology, and long-lived civilizations arise.
  • Small changes in these uncertain factors can change the estimate from almost zero to millions of civilizations.
  • The article gives an illustrative modern estimate of about 46 communicating civilizations, but this should not be treated as a measured value.
  • The Drake equation connects naturally to the Fermi paradox: if intelligent civilizations may be common, why have we found no convincing evidence of them?

ARTICLE / ARTICLE [PDF]]]>
Jørgen Veisdal’s article explains the Drake equation, proposed by astronomer Frank Drake in 1961 as a framework for estimating how many technologically advanced extraterrestrial civilizations might currently be detectable in the Milky Way. The equation is
$N = R_* \times f_p \times n_e \times f_l \times f_i \times f_c \times L$,
where $R_*$ is the rate of star formation, $f_p$ the fraction of stars with planets, $n_e$ the number of potentially habitable planets per planetary system, $f_l$ the fraction on which life develops, $f_i$ the fraction that develops intelligence, $f_c$ the fraction producing detectable communications, and $L$ the duration for which such civilizations remain detectable. The equation was never intended to give a precise answer; rather, Drake devised it to organize the major scientific uncertainties involved in the search for extraterrestrial intelligence. Early optimistic assumptions produced estimates ranging from roughly 20 to tens of millions of communicating civilizations, illustrating how sensitive the result is to the chosen parameters.
 
Astronomical observations have since improved some of these estimates. We now know that planets appear to be extremely common, and studies using Kepler data have suggested that the Milky Way may contain billions of roughly Earth-sized planets in habitable zones. However, the biological and sociological factors—$f_l$, $f_i$, $f_c$, and especially $L$—remain largely speculative because Earth provides essentially our only example of life and technological civilization. Using more modern astronomical estimates while assigning intermediate values such as $0.5$ to the unknown factors, the article obtains an illustrative estimate of about 46 communicating extraterrestrial civilizations in the Milky Way. The author stresses that this number should not be interpreted as a scientific measurement; it mainly demonstrates how assumptions propagate through the Drake equation.
 
The article concludes by connecting the Drake equation with the Fermi paradox: if potentially habitable planets are abundant and technological civilizations are reasonably common, why have we seen no convincing evidence of them? Given the age and enormous number of stars in the Galaxy, even relatively slow interstellar expansion might allow a civilization to spread across the Milky Way in a few million years. Thus the tension between large Drake-equation estimates and the absence of observable extraterrestrials leads to Fermi’s famous question: “Where is everybody?”

Key takeaways:
  • The Drake equation is a framework for estimating the number of detectable extraterrestrial civilizations in the Milky Way.
  • It is not a precise prediction; its value depends heavily on uncertain assumptions.
  • Astronomy has improved estimates for factors such as the number of stars, planets, and potentially habitable worlds.
  • The biggest uncertainties remain biological and technological: how often life, intelligence, communication technology, and long-lived civilizations arise.
  • Small changes in these uncertain factors can change the estimate from almost zero to millions of civilizations.
  • The article gives an illustrative modern estimate of about 46 communicating civilizations, but this should not be treated as a measured value.
  • The Drake equation connects naturally to the Fermi paradox: if intelligent civilizations may be common, why have we found no convincing evidence of them?

ARTICLE / ARTICLE [PDF]]]> <![CDATA[Reflections on the Millennium Problems]]> https://mklab.gr/showthread.php?tid=1912 Tue, 08 Sep 2026 22:04:13 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1912 “Reflections on the Millennium Problems” (August 2026) argues that three famous Millennium Prize Problems—the Riemann Hypothesis, $P$ vs. $NP$, and the existence and smoothness of Navier–Stokes solutions—remain mathematically profound but have gradually become less consequential in the direct practical senses originally associated with them. 

For the Riemann Hypothesis, trillions of computed zeros of $\zeta(s)$ lie on the critical line $\operatorname{Re}(s)=\tfrac12$, so even if RH eventually fails, Trefethen argues that the resulting quantitative disturbance in the observed distribution of primes would probably be extremely small; the greater importance of a proof would be the new mathematical techniques it might unlock, particularly for the Generalized Riemann Hypothesis.

 For $P$ vs. $NP$, decades of computation have shown that worst-case exponential complexity often does not prevent difficult problems from being solved effectively in practice through heuristics, approximation algorithms, special structure, and favorable typical cases, so the distinction has become more of a fundamental organizing principle of theoretical computer science than an absolute practical barrier.

 For Navier–Stokes, increasing evidence suggests that finite-time singularities, if they exist at all, may require highly artificial and unstable initial configurations, making them unlikely to matter for ordinary physical fluid flows. Trefethen’s broader thesis is that extremely long-lived open problems may gradually lose some of their original practical “leverage” precisely because mathematics and computation learn to work successfully around them; nevertheless, their eventual solution would still be historic, mainly because of the new theories and methods developed in pursuing them. 

ARTICLE [PDF]]]> “Reflections on the Millennium Problems” (August 2026) argues that three famous Millennium Prize Problems—the Riemann Hypothesis, $P$ vs. $NP$, and the existence and smoothness of Navier–Stokes solutions—remain mathematically profound but have gradually become less consequential in the direct practical senses originally associated with them. 

For the Riemann Hypothesis, trillions of computed zeros of $\zeta(s)$ lie on the critical line $\operatorname{Re}(s)=\tfrac12$, so even if RH eventually fails, Trefethen argues that the resulting quantitative disturbance in the observed distribution of primes would probably be extremely small; the greater importance of a proof would be the new mathematical techniques it might unlock, particularly for the Generalized Riemann Hypothesis.

 For $P$ vs. $NP$, decades of computation have shown that worst-case exponential complexity often does not prevent difficult problems from being solved effectively in practice through heuristics, approximation algorithms, special structure, and favorable typical cases, so the distinction has become more of a fundamental organizing principle of theoretical computer science than an absolute practical barrier.

 For Navier–Stokes, increasing evidence suggests that finite-time singularities, if they exist at all, may require highly artificial and unstable initial configurations, making them unlikely to matter for ordinary physical fluid flows. Trefethen’s broader thesis is that extremely long-lived open problems may gradually lose some of their original practical “leverage” precisely because mathematics and computation learn to work successfully around them; nevertheless, their eventual solution would still be historic, mainly because of the new theories and methods developed in pursuing them. 

ARTICLE [PDF]]]> <![CDATA[Interested in foundations of mathematics]]> https://mklab.gr/showthread.php?tid=1886 Mon, 07 Sep 2026 23:00:58 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1886 Author: Vladimir Voevodsky
Lecture: ASC 2014, Nanyang Technological University, Singapore, 25 August 2014


Vladimir Voevodsky explains how his experience with increasingly complicated mathematical proofs led him toward computer-assisted proof verification and new foundations of mathematics. He contrasts problems such as solving a Rubik’s Cube—where a solution can be checked directly—with advanced mathematics, where even a published proof may contain hidden errors. He recounts his proof of the Milnor Conjecture, which took years to turn from the central idea into a complete rigorous argument, and then describes a more troubling example: a 1991 theorem he proved with Mikhail Kapranov concerning $\infty$-groupoids. Carlos Simpson later produced a counterexample, and in 2013 Voevodsky finally recognized that not merely the proof but the theorem itself, with their chosen definition, was false. This episode convinced him that relying solely on human checking becomes increasingly risky as mathematics grows more complicated. 

Voevodsky therefore asks whether mathematical proofs could be checked by computers in much the same way that symbolic computation verifies algebraic identities. To accomplish this, both mathematical statements and proofs must be encoded as symbolic objects—a process called formalization—so that software can verify mechanically that a proposed proof genuinely establishes its theorem. He argues that the traditional foundation of mathematics, ZFC set theory, was developed long before computers and was not designed for convenient formalization of modern mathematics. His search for a more suitable foundation led him beginning around 2006 to Univalent Foundations, connecting Martin-Löf type theory, homotopy theory, and foundations. By 2010 he believed a practical system was emerging, and the 2012–13 IAS program helped develop what became Homotopy Type Theory (HoTT). 

The broader aim is not simply to eliminate mistakes but to change how mathematicians work. Voevodsky suggests that the increasing complexity of modern mathematics makes researchers spend more time checking proofs and can make them less willing to pursue daring ideas. Reliable formal verification could transfer part of that checking burden to computers, allowing mathematicians to concentrate more on concepts and discovery. Projects such as UniMath, implemented using the Coq proof assistant, represented early steps toward this vision: large libraries in which definitions, theorems, and proofs can be verified mechanically. 

Key takeaways
  • Mathematical proofs can remain wrong for years, even when written by leading mathematicians and published in respected journals.
  • Formalization converts statements and proofs into symbolic expressions that a computer can mechanically verify.
  • Voevodsky regarded traditional ZFC foundations as poorly adapted to large-scale computer formalization, motivating his development of Univalent Foundations.
  • Homotopy Type Theory and UniMath emerged from this program, combining type theory, topology, and proof assistants in an effort to make computer-verified mathematics practical. 

ARTICLE [PDF]]]> Author: Vladimir Voevodsky
Lecture: ASC 2014, Nanyang Technological University, Singapore, 25 August 2014


Vladimir Voevodsky explains how his experience with increasingly complicated mathematical proofs led him toward computer-assisted proof verification and new foundations of mathematics. He contrasts problems such as solving a Rubik’s Cube—where a solution can be checked directly—with advanced mathematics, where even a published proof may contain hidden errors. He recounts his proof of the Milnor Conjecture, which took years to turn from the central idea into a complete rigorous argument, and then describes a more troubling example: a 1991 theorem he proved with Mikhail Kapranov concerning $\infty$-groupoids. Carlos Simpson later produced a counterexample, and in 2013 Voevodsky finally recognized that not merely the proof but the theorem itself, with their chosen definition, was false. This episode convinced him that relying solely on human checking becomes increasingly risky as mathematics grows more complicated. 

Voevodsky therefore asks whether mathematical proofs could be checked by computers in much the same way that symbolic computation verifies algebraic identities. To accomplish this, both mathematical statements and proofs must be encoded as symbolic objects—a process called formalization—so that software can verify mechanically that a proposed proof genuinely establishes its theorem. He argues that the traditional foundation of mathematics, ZFC set theory, was developed long before computers and was not designed for convenient formalization of modern mathematics. His search for a more suitable foundation led him beginning around 2006 to Univalent Foundations, connecting Martin-Löf type theory, homotopy theory, and foundations. By 2010 he believed a practical system was emerging, and the 2012–13 IAS program helped develop what became Homotopy Type Theory (HoTT). 

The broader aim is not simply to eliminate mistakes but to change how mathematicians work. Voevodsky suggests that the increasing complexity of modern mathematics makes researchers spend more time checking proofs and can make them less willing to pursue daring ideas. Reliable formal verification could transfer part of that checking burden to computers, allowing mathematicians to concentrate more on concepts and discovery. Projects such as UniMath, implemented using the Coq proof assistant, represented early steps toward this vision: large libraries in which definitions, theorems, and proofs can be verified mechanically. 

Key takeaways
  • Mathematical proofs can remain wrong for years, even when written by leading mathematicians and published in respected journals.
  • Formalization converts statements and proofs into symbolic expressions that a computer can mechanically verify.
  • Voevodsky regarded traditional ZFC foundations as poorly adapted to large-scale computer formalization, motivating his development of Univalent Foundations.
  • Homotopy Type Theory and UniMath emerged from this program, combining type theory, topology, and proof assistants in an effort to make computer-verified mathematics practical. 

ARTICLE [PDF]]]> <![CDATA[The Evolving Foundations of Math]]> https://mklab.gr/showthread.php?tid=1868 Sun, 06 Sep 2026 00:53:53 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1868 “The Evolving Foundations of Math” examines how the basic assumptions underlying mathematics have repeatedly been questioned, repaired and rebuilt from the late 19th century to the present. Beginning with Cantor’s revolutionary work on infinity and the development of rigorous set theory, the series traces the tension between intuition, formal proof and logical consistency through figures such as Cantor, Zermelo, Gödel and Grothendieck. It explores controversial alternatives such as ultrafinitism, the paradoxes created by infinity, the growing use of computer-verified proofs, and modern attempts to reconstruct areas such as topology and geometry from more fundamental concepts. The central message is that mathematics is not built on an immutable foundation: its definitions, axioms, methods of proof and even ideas about what mathematical objects should exist continue to evolve as mathematicians encounter paradoxes, new technologies and deeper structural questions. 

Articles in the series

Chapter 1 — Original Sin
  1. Long-Lost Letters Reveal the Lies That Gave Rise To Modern Math — Joseph Howlett
    Cantor, the discovery of different sizes of infinity, and newly uncovered evidence concerning the origins of his famous 1874 result.
  2. How Can Infinity Come in Many Sizes? A Visual Investigation — Mark Belan & Jordana Cepelewicz
    A visual explanation of countable and uncountable infinities.
  3. ‘Fantastic’ Proof Implies the Existence of In-Between Infinities — Natalie Wolchover
    New developments surrounding the continuum problem and possible intermediate sizes of infinity. 

Chapter 2 — Logic Versus Proof
4. In Math, Rigor Is Vital. But Are Digitized Proofs Taking It Too Far? — Leila Sloman
The history of mathematical rigor and the modern movement toward formalizing proofs with systems such as Lean.
5. How Writing Changes Mathematical Thought — John Pavlus
An interview exploring how mathematical notation influences the way mathematicians think.
6. The Jagged, Monstrous Function That Broke Calculus — Solomon Adams
How pathological functions challenged 19th-century intuition and forced mathematicians to make calculus more rigorous. 

Chapter 3 — Cut to the Core
7. Mathematicians Want to Banish Infinity. What Might They Gain? — Gregory Barber
An exploration of ultrafinitism, which questions whether infinite mathematical objects should exist at all.
8. Why Math’s Final Axiom Proved So Controversial — Gregory Barber
The difficult historical development of Zermelo-Fraenkel set theory and its axioms.
9. Gödel’s Incompleteness Proof, Explained — Natalie Wolchover
An accessible account of Gödel’s incompleteness theorems and the fundamental limits they place on formal mathematical systems.
10. How Infinity Leads to One of Math’s Strangest Paradoxes — Max G. Levy
The Banach-Tarski paradox and the counterintuitive consequences of infinity. 

Chapter 4 — A Revolution Begins
11. Two Researchers Are Rebuilding Mathematics From the Ground Up — Konstantin Kakaes
Peter Scholze and Dustin Clausen’s ambitious program to rethink fundamental concepts in topology and number theory.
12. How Alexander Grothendieck Revolutionized 20th-Century Mathematics — Konstantin Kakaes
An overview of Grothendieck’s transformative ideas in algebraic geometry.
13. Inside the Fight to Fix Geometry’s Foundations — Kevin Hartnett
A debate over whether fundamental arguments in modern geometry can truly be trusted and how they should be rebuilt.

Overall, the collection moves roughly from Cantor and infinity → rigor and formal proof → axioms and Gödel → Grothendieck and new foundations, making it essentially a compact history of how mathematicians have repeatedly reconsidered what mathematics is built from and what counts as a valid proof.

ARTICLES]]> “The Evolving Foundations of Math” examines how the basic assumptions underlying mathematics have repeatedly been questioned, repaired and rebuilt from the late 19th century to the present. Beginning with Cantor’s revolutionary work on infinity and the development of rigorous set theory, the series traces the tension between intuition, formal proof and logical consistency through figures such as Cantor, Zermelo, Gödel and Grothendieck. It explores controversial alternatives such as ultrafinitism, the paradoxes created by infinity, the growing use of computer-verified proofs, and modern attempts to reconstruct areas such as topology and geometry from more fundamental concepts. The central message is that mathematics is not built on an immutable foundation: its definitions, axioms, methods of proof and even ideas about what mathematical objects should exist continue to evolve as mathematicians encounter paradoxes, new technologies and deeper structural questions. 

Articles in the series

Chapter 1 — Original Sin
  1. Long-Lost Letters Reveal the Lies That Gave Rise To Modern Math — Joseph Howlett
    Cantor, the discovery of different sizes of infinity, and newly uncovered evidence concerning the origins of his famous 1874 result.
  2. How Can Infinity Come in Many Sizes? A Visual Investigation — Mark Belan & Jordana Cepelewicz
    A visual explanation of countable and uncountable infinities.
  3. ‘Fantastic’ Proof Implies the Existence of In-Between Infinities — Natalie Wolchover
    New developments surrounding the continuum problem and possible intermediate sizes of infinity. 

Chapter 2 — Logic Versus Proof
4. In Math, Rigor Is Vital. But Are Digitized Proofs Taking It Too Far? — Leila Sloman
The history of mathematical rigor and the modern movement toward formalizing proofs with systems such as Lean.
5. How Writing Changes Mathematical Thought — John Pavlus
An interview exploring how mathematical notation influences the way mathematicians think.
6. The Jagged, Monstrous Function That Broke Calculus — Solomon Adams
How pathological functions challenged 19th-century intuition and forced mathematicians to make calculus more rigorous. 

Chapter 3 — Cut to the Core
7. Mathematicians Want to Banish Infinity. What Might They Gain? — Gregory Barber
An exploration of ultrafinitism, which questions whether infinite mathematical objects should exist at all.
8. Why Math’s Final Axiom Proved So Controversial — Gregory Barber
The difficult historical development of Zermelo-Fraenkel set theory and its axioms.
9. Gödel’s Incompleteness Proof, Explained — Natalie Wolchover
An accessible account of Gödel’s incompleteness theorems and the fundamental limits they place on formal mathematical systems.
10. How Infinity Leads to One of Math’s Strangest Paradoxes — Max G. Levy
The Banach-Tarski paradox and the counterintuitive consequences of infinity. 

Chapter 4 — A Revolution Begins
11. Two Researchers Are Rebuilding Mathematics From the Ground Up — Konstantin Kakaes
Peter Scholze and Dustin Clausen’s ambitious program to rethink fundamental concepts in topology and number theory.
12. How Alexander Grothendieck Revolutionized 20th-Century Mathematics — Konstantin Kakaes
An overview of Grothendieck’s transformative ideas in algebraic geometry.
13. Inside the Fight to Fix Geometry’s Foundations — Kevin Hartnett
A debate over whether fundamental arguments in modern geometry can truly be trusted and how they should be rebuilt.

Overall, the collection moves roughly from Cantor and infinity → rigor and formal proof → axioms and Gödel → Grothendieck and new foundations, making it essentially a compact history of how mathematicians have repeatedly reconsidered what mathematics is built from and what counts as a valid proof.

ARTICLES]]> <![CDATA[Mathematical coincidence]]> https://mklab.gr/showthread.php?tid=1796 Thu, 03 Sep 2026 02:10:37 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1796
A mathematical coincidence occurs when two apparently unrelated mathematical expressions have values that are unexpectedly close, without an obvious theoretical reason. A simple example is
$2^{10}=1024\approx1000=10^3$.
Such coincidences often involve famous constants such as $\pi$, $e$, the golden ratio $\varphi$, or simple integers. Some arise because a rational number happens to approximate an irrational number extremely well—for example, $355/113$ approximates $\pi$ to six decimal places. Continued fractions can explain why some of these approximations are so accurate, although the deeper reason that unusually good approximations occur is not always clear. Other striking examples include $\pi^2\approx10$, $\pi^3\approx31$, and $2\pi+e\approx9$. 

Importantly, something that looks like a coincidence may eventually turn out to have a genuine mathematical explanation. A famous example is $e^\pi-\pi\approx20$, whose accuracy can be connected to identities involving the Jacobi theta function. Even more spectacular is Ramanujan's constant, $e^{\pi\sqrt{163}}$, which is extraordinarily close to an integer; this is not simply an accident but is connected with the special number-theoretic properties of the Heegner number $163$. The article also discusses coincidences in physics and measurement—for example, the speed of light being close to $3\times10^8$ m/s, Earth's gravitational acceleration being close to $10$ m/s², and the apparent sizes of the Sun and Moon being similar enough to permit total solar eclipses. Some of these depend partly on how humans defined measurement units, while others are genuinely accidental numerical relationships.

Key takeaways
  • Mathematical coincidences are unexpected near-equalities between seemingly unrelated quantities.
  • Some are genuinely accidental, while others hide deeper mathematics involving continued fractions, number theory, modular functions, or special constants.
  • Famous examples include $2^{10}\approx10^3$, $\pi^2\approx10$, $2\pi+e\approx9$, and $e^{\pi\sqrt{163}}\approx$ an integer.
  • Their main appeal is mathematical curiosity, but some coincidences also provide useful engineering approximations or occasionally lead mathematicians toward deeper structures. 

ARTICLE]]>
A mathematical coincidence occurs when two apparently unrelated mathematical expressions have values that are unexpectedly close, without an obvious theoretical reason. A simple example is
$2^{10}=1024\approx1000=10^3$.
Such coincidences often involve famous constants such as $\pi$, $e$, the golden ratio $\varphi$, or simple integers. Some arise because a rational number happens to approximate an irrational number extremely well—for example, $355/113$ approximates $\pi$ to six decimal places. Continued fractions can explain why some of these approximations are so accurate, although the deeper reason that unusually good approximations occur is not always clear. Other striking examples include $\pi^2\approx10$, $\pi^3\approx31$, and $2\pi+e\approx9$. 

Importantly, something that looks like a coincidence may eventually turn out to have a genuine mathematical explanation. A famous example is $e^\pi-\pi\approx20$, whose accuracy can be connected to identities involving the Jacobi theta function. Even more spectacular is Ramanujan's constant, $e^{\pi\sqrt{163}}$, which is extraordinarily close to an integer; this is not simply an accident but is connected with the special number-theoretic properties of the Heegner number $163$. The article also discusses coincidences in physics and measurement—for example, the speed of light being close to $3\times10^8$ m/s, Earth's gravitational acceleration being close to $10$ m/s², and the apparent sizes of the Sun and Moon being similar enough to permit total solar eclipses. Some of these depend partly on how humans defined measurement units, while others are genuinely accidental numerical relationships.

Key takeaways
  • Mathematical coincidences are unexpected near-equalities between seemingly unrelated quantities.
  • Some are genuinely accidental, while others hide deeper mathematics involving continued fractions, number theory, modular functions, or special constants.
  • Famous examples include $2^{10}\approx10^3$, $\pi^2\approx10$, $2\pi+e\approx9$, and $e^{\pi\sqrt{163}}\approx$ an integer.
  • Their main appeal is mathematical curiosity, but some coincidences also provide useful engineering approximations or occasionally lead mathematicians toward deeper structures. 

ARTICLE]]> <![CDATA[The Women Taking Math To The Next Dimension]]> https://mklab.gr/showthread.php?tid=1785 Wed, 02 Sep 2026 01:12:02 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1785 Author: Lauren J. Young
Published: November 16, 2017 — Science Friday 

The article profiles mathematicians Rebecca Goldin, Emily Riehl, and Eugenia Cheng, who challenge the common perception of mathematics as little more than calculations and formulas. They describe mathematics instead as a highly creative discipline driven by imagination, abstraction, problem solving, and the search for underlying structures. Goldin works on geometry and symmetry, Riehl studies higher-dimensional category theory, and Cheng works in category theory, which she describes as the “mathematics of mathematics.” Their examples range from the Fibonacci sequence and the Banach–Tarski paradox to non-Euclidean geometry, showing how mathematical ideas can lead far beyond what can be physically visualized. 

A recurring theme is that doing mathematics is much more social and exploratory than outsiders often imagine. Riehl emphasizes that research frequently involves collaboration and that asking questions is essential rather than a sign of weakness. Goldin says that she originally expected mathematics to become increasingly tedious, but instead discovered that advanced mathematics is playful, conceptual, and visually beautiful. Cheng similarly stresses that mathematics combines rigorous logic with creativity and imagination: because mathematicians are not restricted by physical reality, they can investigate structures of almost unlimited complexity. 

The three mathematicians also offer advice to young women considering mathematical careers. Goldin encourages them to pursue mathematics without feeling they must already have a career mapped out; Riehl urges students to ask questions freely; and Cheng argues that women do not need to imitate stereotypically male behavior in order to succeed. Overall, the article presents mathematics not as a cold collection of computations but as an imaginative, collaborative intellectual activity in which curiosity and creativity are as important as technical ability.

Key takeaways
  • Advanced mathematics is fundamentally about ideas, structures, patterns, and imagination, not merely calculation.
  • Mathematical research can be highly collaborative and social.
  • Understanding difficult mathematics often develops gradually over years rather than arriving instantly.
  • Asking questions is an important part of becoming a better mathematician.
  • The article argues against stereotypes about both what mathematics is and who can become a mathematician.

ARTICLE]]> Author: Lauren J. Young
Published: November 16, 2017 — Science Friday 

The article profiles mathematicians Rebecca Goldin, Emily Riehl, and Eugenia Cheng, who challenge the common perception of mathematics as little more than calculations and formulas. They describe mathematics instead as a highly creative discipline driven by imagination, abstraction, problem solving, and the search for underlying structures. Goldin works on geometry and symmetry, Riehl studies higher-dimensional category theory, and Cheng works in category theory, which she describes as the “mathematics of mathematics.” Their examples range from the Fibonacci sequence and the Banach–Tarski paradox to non-Euclidean geometry, showing how mathematical ideas can lead far beyond what can be physically visualized. 

A recurring theme is that doing mathematics is much more social and exploratory than outsiders often imagine. Riehl emphasizes that research frequently involves collaboration and that asking questions is essential rather than a sign of weakness. Goldin says that she originally expected mathematics to become increasingly tedious, but instead discovered that advanced mathematics is playful, conceptual, and visually beautiful. Cheng similarly stresses that mathematics combines rigorous logic with creativity and imagination: because mathematicians are not restricted by physical reality, they can investigate structures of almost unlimited complexity. 

The three mathematicians also offer advice to young women considering mathematical careers. Goldin encourages them to pursue mathematics without feeling they must already have a career mapped out; Riehl urges students to ask questions freely; and Cheng argues that women do not need to imitate stereotypically male behavior in order to succeed. Overall, the article presents mathematics not as a cold collection of computations but as an imaginative, collaborative intellectual activity in which curiosity and creativity are as important as technical ability.

Key takeaways
  • Advanced mathematics is fundamentally about ideas, structures, patterns, and imagination, not merely calculation.
  • Mathematical research can be highly collaborative and social.
  • Understanding difficult mathematics often develops gradually over years rather than arriving instantly.
  • Asking questions is an important part of becoming a better mathematician.
  • The article argues against stereotypes about both what mathematics is and who can become a mathematician.

ARTICLE]]> <![CDATA[Solving the Flat Cube]]> https://mklab.gr/showthread.php?tid=1740 Fri, 28 Aug 2026 18:30:53 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1740 Published: August 19, 2026

James Propp introduces the Flat Cube, a planar puzzle inspired by the Rubik’s Cube. Instead of small cubes, it consists of lozenges (rhombi) tiling a hexagon. Whenever three lozenges form a small hexagon, they may be rotated around its center in multiples of $60^\circ$. The goal is to restore a scrambled configuration using the fewest possible twists. This leads to an analogue of the Rubik’s Cube’s “God’s number”: the maximum, over all possible scrambled states, of the minimum number of moves required to solve the puzzle. Propp initially proves that this number must be at least 27. 

He gives two elegant visual proofs. The first adds a dimension: a lozenge tiling can be interpreted as the visible surface of a pile of unit cubes inside a $3\times3\times3$ box. A legal twist corresponds to adding or removing exactly one cube. Transforming the empty configuration into the completely filled one therefore requires at least
33=273^3=27
moves. The second proof removes a dimension: considering only the nine horizontally oriented lozenges turns them into beads sliding along one-dimensional tracks. Each of the $9$ beads must move $3$ positions, while one twist moves only one bead by one position, giving again
9×3=27.9\times3=27.
For the uncolored version, this reasoning can in fact be pushed further to show that the worst-case distance is exactly $27$. 

The article also connects the puzzle with the mathematics of lozenge tilings, plane partitions, combinatorics, and configuration spaces. Propp collaborated with puzzle designer Oskar van Deventer and Dmitry Andreev to turn the mathematical idea into an actual mechanical puzzle, called the Propp Twist. Particularly interesting are updates added two days after publication: Tom Rokicki showed that the order-$2$ Flat Cube has God’s number exactly $27$, while two arbitrary states can be as far as $30$ moves apart. More strikingly, Rui Viana subsequently proved that for the order-$3$ Flat Cube, God’s number is at least $81$, and Rokicki found a pair of states separated by 91 moves. Thus the problem is substantially richer than the original $27$-move argument suggests. 

Key takeaways
  • The Flat Cube converts a geometric tiling problem into a Rubik-like optimization problem.
  • A lozenge rotation can be interpreted either as adding/removing a cube or as moving a bead one step.
  • Both viewpoints give the beautiful lower-bound calculation $27=3^3=9\times3$.
  • The later results show that the full colored order-$3$ puzzle is considerably harder: its God’s number is already known to be at least $81$, with some pairs of states 91 moves apart. 

ARTICLE [PDF]]]> Published: August 19, 2026

James Propp introduces the Flat Cube, a planar puzzle inspired by the Rubik’s Cube. Instead of small cubes, it consists of lozenges (rhombi) tiling a hexagon. Whenever three lozenges form a small hexagon, they may be rotated around its center in multiples of $60^\circ$. The goal is to restore a scrambled configuration using the fewest possible twists. This leads to an analogue of the Rubik’s Cube’s “God’s number”: the maximum, over all possible scrambled states, of the minimum number of moves required to solve the puzzle. Propp initially proves that this number must be at least 27. 

He gives two elegant visual proofs. The first adds a dimension: a lozenge tiling can be interpreted as the visible surface of a pile of unit cubes inside a $3\times3\times3$ box. A legal twist corresponds to adding or removing exactly one cube. Transforming the empty configuration into the completely filled one therefore requires at least
33=273^3=27
moves. The second proof removes a dimension: considering only the nine horizontally oriented lozenges turns them into beads sliding along one-dimensional tracks. Each of the $9$ beads must move $3$ positions, while one twist moves only one bead by one position, giving again
9×3=27.9\times3=27.
For the uncolored version, this reasoning can in fact be pushed further to show that the worst-case distance is exactly $27$. 

The article also connects the puzzle with the mathematics of lozenge tilings, plane partitions, combinatorics, and configuration spaces. Propp collaborated with puzzle designer Oskar van Deventer and Dmitry Andreev to turn the mathematical idea into an actual mechanical puzzle, called the Propp Twist. Particularly interesting are updates added two days after publication: Tom Rokicki showed that the order-$2$ Flat Cube has God’s number exactly $27$, while two arbitrary states can be as far as $30$ moves apart. More strikingly, Rui Viana subsequently proved that for the order-$3$ Flat Cube, God’s number is at least $81$, and Rokicki found a pair of states separated by 91 moves. Thus the problem is substantially richer than the original $27$-move argument suggests. 

Key takeaways
  • The Flat Cube converts a geometric tiling problem into a Rubik-like optimization problem.
  • A lozenge rotation can be interpreted either as adding/removing a cube or as moving a bead one step.
  • Both viewpoints give the beautiful lower-bound calculation $27=3^3=9\times3$.
  • The later results show that the full colored order-$3$ puzzle is considerably harder: its God’s number is already known to be at least $81$, with some pairs of states 91 moves apart. 

ARTICLE [PDF]]]> <![CDATA[The Math Ceiling]]> https://mklab.gr/showthread.php?tid=1713 Thu, 20 Aug 2026 19:44:30 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1713 The Math Ceiling: Where’s Your Cognitive Breaking Point?
Author: Ben Orlin
Published: April 8, 2015
Source:Math with Bad Drawings 

Ben Orlin examines the common belief that every person has a natural “mathematical ceiling”—a point beyond which their cognitive ability simply cannot carry them. He contrasts two attitudes: the traditional view that some people are inherently “math people” while others are not, and the more optimistic educational view that essentially anyone can learn mathematics given sufficient effort. Orlin proposes a third explanation: many apparent mathematical ceilings may not be biological limits at all, but the accumulated consequences of gaps in earlier understanding.

He illustrates the idea with what he calls the “Law of the Broken Futon.” A futon with one small structural piece missing may initially appear perfectly usable, but as increasing weight is placed upon it, the unsupported parts gradually deform until the whole structure collapses. Mathematical learning can work the same way. A student may successfully manipulate linear equations, for example, without understanding that a graph represents all ordered pairs satisfying an equation. Procedures and memorized rules can compensate temporarily, but later topics—quadratics, trigonometry, calculus—place greater conceptual weight on that missing foundation. By then, repairing the gap is harder because the learner has constructed years of shortcuts around it. 

Orlin therefore argues that what students experience as their “cognitive breaking point” may often be educationally created rather than intellectually predetermined. Teaching students algorithms without understanding can allow them to pass exams and advance through the curriculum, while silently weakening the structure on which later mathematics depends. His central warning is especially striking: a learner who repeatedly gets correct answers without knowing why the methods work may appear successful today but is being prepared for difficulty tomorrow. 

Key takeaways
  • Mathematical difficulty is cumulative: small conceptual gaps can become major obstacles years later.
  • Procedural competence is not the same as understanding. Being able to execute an algorithm can conceal serious weaknesses.
  • Many supposed “math ceilings” may actually be consequences of earlier teaching and learning, rather than fixed limits of intelligence.
  • The educational priority should therefore be deep conceptual understanding before progression, even when this means moving more slowly through the curriculum. 

ARTICLE]]> The Math Ceiling: Where’s Your Cognitive Breaking Point?
Author: Ben Orlin
Published: April 8, 2015
Source:Math with Bad Drawings 

Ben Orlin examines the common belief that every person has a natural “mathematical ceiling”—a point beyond which their cognitive ability simply cannot carry them. He contrasts two attitudes: the traditional view that some people are inherently “math people” while others are not, and the more optimistic educational view that essentially anyone can learn mathematics given sufficient effort. Orlin proposes a third explanation: many apparent mathematical ceilings may not be biological limits at all, but the accumulated consequences of gaps in earlier understanding.

He illustrates the idea with what he calls the “Law of the Broken Futon.” A futon with one small structural piece missing may initially appear perfectly usable, but as increasing weight is placed upon it, the unsupported parts gradually deform until the whole structure collapses. Mathematical learning can work the same way. A student may successfully manipulate linear equations, for example, without understanding that a graph represents all ordered pairs satisfying an equation. Procedures and memorized rules can compensate temporarily, but later topics—quadratics, trigonometry, calculus—place greater conceptual weight on that missing foundation. By then, repairing the gap is harder because the learner has constructed years of shortcuts around it. 

Orlin therefore argues that what students experience as their “cognitive breaking point” may often be educationally created rather than intellectually predetermined. Teaching students algorithms without understanding can allow them to pass exams and advance through the curriculum, while silently weakening the structure on which later mathematics depends. His central warning is especially striking: a learner who repeatedly gets correct answers without knowing why the methods work may appear successful today but is being prepared for difficulty tomorrow. 

Key takeaways
  • Mathematical difficulty is cumulative: small conceptual gaps can become major obstacles years later.
  • Procedural competence is not the same as understanding. Being able to execute an algorithm can conceal serious weaknesses.
  • Many supposed “math ceilings” may actually be consequences of earlier teaching and learning, rather than fixed limits of intelligence.
  • The educational priority should therefore be deep conceptual understanding before progression, even when this means moving more slowly through the curriculum. 

ARTICLE]]> <![CDATA[The Reasonable Ineffectiveness of Mathematics]]> https://mklab.gr/showthread.php?tid=1708 Thu, 20 Aug 2026 19:01:53 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1708 Author: Derek Abbott
Published: October 2013
Journal:Proceedings of the IEEE, Vol. 101, No. 10, pp. 2147–2153
DOI: 10.1109/JPROC.2013.2274907 

Derek Abbott challenges Eugene Wigner’s famous claim about the “unreasonable effectiveness of mathematics” in describing nature. Rather than regarding the success of mathematics in physics as something mysterious or as evidence that mathematical objects have an independent Platonic existence, Abbott argues that mathematics is largely a human invention constructed to represent patterns and regularities that humans happen to observe. We naturally notice problems that our mathematical tools can solve, develop new mathematics when existing tools fail, and tend to remember successful mathematical models while overlooking the enormous number of unsuccessful ones. Thus, what appears to be a miraculous correspondence between mathematics and nature may partly be a form of selection bias.

Abbott develops Richard Hamming's earlier observations: we see what we are equipped to look for; we choose the mathematics appropriate to particular problems; science successfully addresses only a comparatively small subset of conceivable questions; and human cognition itself was shaped by evolution to understand phenomena occurring on approximately human spatial and temporal scales. Abbott adds two ideas of his own. First, mathematical laws function as a kind of lossy compression of reality: equations discard noise and complexity in order to give the human mind compact, usable descriptions. Second, scientific models undergo something like Darwinian selection—successful models survive and are published, while thousands of unsuccessful ideas disappear, making mathematics appear more consistently successful than it actually is.

He illustrates this with engineering, nonlinear systems, fractals, complex numbers and even the apparently elementary act of counting bananas. In the physical world, objects have fuzzy boundaries, measurements contain noise, and there are ultimately physical limits to how many objects can actually be counted or represented. Mathematical entities such as perfect circles, integers extending indefinitely, delta functions and exact real numbers therefore need not correspond to independently existing objects in nature. Abbott adopts a deliberately strong non-Platonist position: mathematical models should be regarded primarily as useful constructions rather than literal descriptions of what reality “really is.” This view has practical consequences, he argues, because treating mathematics as something humans are free to redesign may encourage better formalisms—for example, his advocacy of geometric algebra as a more unified alternative to the traditional mixture of scalars, vectors, complex numbers, quaternions, dot products and cross products.

Key takeaways
  • Mathematics works, but not miraculously: its successes may look extraordinary because we disproportionately notice successful applications.
  • Mathematical models are approximations: Abbott describes them as compressed, idealized representations of a noisy and complicated reality.
  • Mathematics may be anthropocentric: our mathematics reflects human perception, evolution, cognitive limitations and the scales at which we live.
  • The article argues against mathematical Platonism: Abbott concludes that mathematics is best viewed as a powerful human-created tool for describing regularities, not necessarily as a pre-existing structure of the universe.

ARTICLE [PDF]]]> Author: Derek Abbott
Published: October 2013
Journal:Proceedings of the IEEE, Vol. 101, No. 10, pp. 2147–2153
DOI: 10.1109/JPROC.2013.2274907 

Derek Abbott challenges Eugene Wigner’s famous claim about the “unreasonable effectiveness of mathematics” in describing nature. Rather than regarding the success of mathematics in physics as something mysterious or as evidence that mathematical objects have an independent Platonic existence, Abbott argues that mathematics is largely a human invention constructed to represent patterns and regularities that humans happen to observe. We naturally notice problems that our mathematical tools can solve, develop new mathematics when existing tools fail, and tend to remember successful mathematical models while overlooking the enormous number of unsuccessful ones. Thus, what appears to be a miraculous correspondence between mathematics and nature may partly be a form of selection bias.

Abbott develops Richard Hamming's earlier observations: we see what we are equipped to look for; we choose the mathematics appropriate to particular problems; science successfully addresses only a comparatively small subset of conceivable questions; and human cognition itself was shaped by evolution to understand phenomena occurring on approximately human spatial and temporal scales. Abbott adds two ideas of his own. First, mathematical laws function as a kind of lossy compression of reality: equations discard noise and complexity in order to give the human mind compact, usable descriptions. Second, scientific models undergo something like Darwinian selection—successful models survive and are published, while thousands of unsuccessful ideas disappear, making mathematics appear more consistently successful than it actually is.

He illustrates this with engineering, nonlinear systems, fractals, complex numbers and even the apparently elementary act of counting bananas. In the physical world, objects have fuzzy boundaries, measurements contain noise, and there are ultimately physical limits to how many objects can actually be counted or represented. Mathematical entities such as perfect circles, integers extending indefinitely, delta functions and exact real numbers therefore need not correspond to independently existing objects in nature. Abbott adopts a deliberately strong non-Platonist position: mathematical models should be regarded primarily as useful constructions rather than literal descriptions of what reality “really is.” This view has practical consequences, he argues, because treating mathematics as something humans are free to redesign may encourage better formalisms—for example, his advocacy of geometric algebra as a more unified alternative to the traditional mixture of scalars, vectors, complex numbers, quaternions, dot products and cross products.

Key takeaways
  • Mathematics works, but not miraculously: its successes may look extraordinary because we disproportionately notice successful applications.
  • Mathematical models are approximations: Abbott describes them as compressed, idealized representations of a noisy and complicated reality.
  • Mathematics may be anthropocentric: our mathematics reflects human perception, evolution, cognitive limitations and the scales at which we live.
  • The article argues against mathematical Platonism: Abbott concludes that mathematics is best viewed as a powerful human-created tool for describing regularities, not necessarily as a pre-existing structure of the universe.

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