Inbjudan till NCM-Seminarium om DigiMat

Tordagen den 19/11 kl 16-17.30 presenterar jag DigiMat vid ett zoom-seminarium hos Nationellt Centrum för Matematik NCM. Gör anmälan för att följa/delta.

DigiMat är nytt läromedel som realiserar den nya läroplanen 2018 med programmering som del av matematikundervisningen i skolan. 

Seminariet vänder sig till alla lärare i matematik/teknik i grundskola och gymnasium. Nästa matematiklektion kan startas tillsammans med eleverna genom ett klick på DigiMat School.

Välkomna!

  

 

DigiMat US Election 2020

Breaking News: DigiMat School is opening 

• US Election Betting  

based on digital mathematics free from interference. Click US Election Betting top left and then arrow for new simulation/prediction.

 

Grundlagsskydd mot Faktaresistens

Leif Levin, prof em i statskunskap, pläderar i dagens DN Debatt Ge forskningen skydd mot antidemokraterna:

• Och inre fiender har börjat under­minera demokratin och fört utvecklingen i auktoritär riktning.
• Allt färre länder visar sig vara oanfrätta. 
• De auktoritära tendenserna är särskilt påfallande i  Kina...och USA.
• Svenska politiker verkar ovilliga att använda grundlagen som skydd mot samtidens antidemokratiska och populistiska framstötar. För någon månad sedan misslyckades riksdagspartierna att enas om att ge public service ett grundlagsskydd mot politisk styrning.
• Jag tycker det vore prydligt om vi då fick höra statsrådet något reflektera över det allvarliga läge som den globala demokratin hamnat i och universitetens möjligheter att bekämpa den faktaresistens som är förknippad med de auktoritära och populistiska tendenserna.

Levin talar inte om vilka som är de inre fienderna, vilka länder som är oanfrätta (Sverige, Andorra?), vilka som är antidemokratiska (utöver Kina och USA?), vilka som lider av faktaresistens (USA och NordKorea?), men antyder att det hela kopplar till populistiska framstötar och tendenser. Vi får själva gissa. Det är försåtligt. 

Men Levin talar också klarspråk: Faktaresistens skall bekämpas med hjälp av Grundlagen. Men det behövs inte: Public service fixar redan detta genom att via politisk styrning tala om för medborgarna vad som är fakta: Lyssna på Vetenskapsradion Klotet! 

Och Leif: Om nu public service skulle ges grundlagsskydd mot politisk styrning, så skulle ju Klotet stoppas  och faktaresistensen få fritt spelrum.  Så vad menar Du egentligen? Och vad är demokrati om inte populism i den meningen att folket, och inte bara eliten, har rätt att uttrycka åsikter (om fakta)? Och vad är den Globala Demokratin? Frågor till innehavaren (1972-2008) av den Skytteanska professuren i vältalighet och statskunskap.

PS Den framstående tänkaren och författaren Sven-Eric Liedman meddelar svenska folket i DN Idé-Debatt (Åsikten får aldrig vara viktigare än insikten) att han kan skilja åsikt från insikt och minsann inte alls är någon dumbom:

• År 2020 gör jag mig till ignorant och dumbom om jag ifrågasätter de människo­skapade klimatförändringarna.
• ...vi som fortfarande har respekt för kunskaper måste uppbåda så mycket motstånd mot den pågående fördumningen som vi bara kan.

Men vilken insikt har Sven-Eric vad gäller dynamiken av jordens klimatförändringar utöver de åsikter som pumpas ut av DN med Greta i ledningen?  

Att Följa Vad Som Predikas av Stat, Skola och Stormedia

Stat, skola och stormedia översköljer medborgarna dagligen och stundligen med ett budskap om den absoluta nödvändigheten av en total omvälvning till ett nytt fossilfritt samhälle, och att det är bråttom att sätta igång nedmonteringen av nuvarande samhälle byggt på fossil energi. 

Under denna massiva propaganda är det inte konstigt att några medborgare tror på budskapets allvar och med förskräckelse ser att ingenting görs. Det gäller Greta och i mer extrem form Andreas Malm som i en stor artikel på 3 sidor i SvD uppmanar till sabotage av fossila energikällor, se SvD Hyllar en Vänsterextremist.

Ju fler Andreas Malm, desto snabbare marsch in i det nya fossilfria samhället.  

Men tidskriften Kvartal har intervjuat en sansad röst. Misstag?

PS DN kontrar SvDs artikel: Greta blir Chefredaktör!

Statsvetare om Staten vs Folket

Professorerna i Statsvetenskap Ulf Bjereld och Marie Demker stryker Staten medhårs i DN Debatt:

• Åsiktskorridorer är en viktig grund för samhällslivet.
• Också i svensk samhällsdebatt ställs kampen om ”sanningen” snarare än kampen om politiska visioner i centrum.
• I en värld där populistiska tankeströmningar växer sig starka riskerar sanningen att reduceras till den verklighetsbeskrivning som bäst motsvarar ”folkets” uppfattning i dess konflikt med samhällets ”elit”.

Så talar professorer i Statsvetenskap betalda av Staten. Demker är ordförande i Socialdemokrater för Tro och Solidaritet och Bjereld tidigare ordf.

What is a 2020 Nobel Physics Prize Singularity?

The motivation 2020 Nobel Physics Prize awarded Roger Penrose reads:

• In January 1965, ten years after Einstein’s death, Roger Penrose proved that black holes really can form and described them in detail; at their heart, black holes hide a singularity in which all the known laws of nature cease. His ground-breaking article is still regarded as the most important contribution to the general theory of relativity since Einstein.
• Penrose proceeded to prove that once a trapped surface had formed, it is impossible, within the theory of general relativity and with a positive energy density, to prevent the collapse towards a singularity (Penrose 1965).

A singularity is thus described as something in which all the known laws of nature cease. 

Let us try to understand what the meaning of such statement can be, and let first recall that mathematics is filled with singularities, and that there are ways to handle such things. As a typical example, take a look att the function $f(x)$ defined by 

• $f(x) = 0$ for $x < 0$
• $f(x) = 1$ for $x > 0$, 

with the value for x = 0 left unspecified. This is a function which "jumps" from 0 to 1 at the point $x = 0$ with a derivative which appears to be infinite. Nothing very strange, although you may recall that

•  Natura non facit saltus.   (Nature does not jump)  (Axiom of Leibniz)

meaning that the physical jump takes place over a certain distance with a large derivative. 

In any case, you may say that the function $f(x)$ "ceases to exist" for x = 0, since no value is specified. But there is something, which is well defined, and that is the jump ( = 1). So there is a form of singularity, where the function f(x) ceases to exist/is not defined, but still the jump of f(x) exists and is well defined.  

There is a well developed mathematical theory to deal with singularities and jumps, which is called distribution theory. In this theory the properties the function f(x) would be expressed through integrals

• $\int f(x)\phi (x) dx$ 

where $\phi (x)$ is any smooth test function. The jump of f(x) would then come out as

• $-\int f(x)\phi^\prime (x)\, dx = \phi (0)$  where $\phi^\prime = \frac{d\phi}{dx}$ is the derivative of $\phi$,

which can be phrased as: The derivative $f^\prime (0)$ is a delta function at 0, which gives a precise meaning to a derivative which appears to be infinite in a standard pointwise sense.  

A (unit) point mass is an elementary concept in mechanics, like a very small ball of very high density with total mass = 1, which can be represented by a delta function.  So a point mass is a singularity without any mystery whatsoever, and it would be futile to ask for any internal physics of a such a thing, or to say that inside a point mass laws of physics cease.

In mathematics singularities thus can have well defined properties (expressing laws of physics), and so a singularity may not be all that mysterious and without any physics!  Why is then the singularity discovered by Penrose worth a Nobel Prize?

Maybe then after all, physics does not cease to exist, because physics cannot cease to exist: The show must go on. (Leibniz).

  

Penrose vs Quantum Mechanics vs Consciousness

2020 Nobel Physics Laurate Roger Penrose claims that 

1. Quantum mechanics is inconsistent (=wrong).
2. Consciousness is governed by quantum mechanics.
3. AI is impossible.
4. There has to be something in physics that we don't yet understand, which is very important, and which is of a non-computational character.

Hopefully Penrose can elaborate on these themes in his Nobel Lecture. 

2020 Physics Nobel Prize Motivation

Roger Penrose is awarded half of the 2020 Nobel Prize in Physics

• for the discovery that black hole formation is a robust prediction of the general theory of relativity.

Let us try to analyse this (seemingly cryptic) motivation. It starts with a discovery of something, but what is that Penrose has discovered? And it ends with a robust prediction of something, but what is that is so robustly predicted? We seek answers in the theoretical foundation presented by the Academy: 

• Penrose proceeded to prove that once a trapped surface had formed, it is impossible, within the theory of general relativity and with a positive energy density, to prevent the collapse towards a singularity (Penrose 1965).
• Penrose’s result is heralded as the first post-Einsteinian result in general relativity.

It appears that what Penrose has proved by logical reasoning within the mathematical general theory of relativity, is that, under some conditions,

• it is impossible to prevent collapse towards a singularity.

Questions: In what sense is this a discovery with physical meaning? Does it is impossible to prevent have a physical meaning? Does collapse towards a singularity have a physical meaning when a singularity has no meaning? Is the robustness connected to the use of logic?  Can a mathematical singularity have some internal physical structure, or is it not a singularity if so? 

Remark 1: The solution of a differential equation can develop a singularity in finite time, like the function x(t) = 1/(1-t), which solves dx/dt = x*x for 0<t<1 with x(0) = 1, and x(t) tends to infinity as t approaches 1 where the solution ceases to exist. But in physics things cannot become infinite and simply cease to exist, and so one can say that the equations dx/dt = x*x cannot describe any physical reality. To discover that 1/(1-t) has a singularity for t = 1 cannot be viewed as a discovery of some physics, only a consequence of some mathematics. 

Remark 2: Einstein's equations have such a cryptic formulation that solutions cannot be found in any form of generality, analytically or computationally. To predict what solutions will do under such conditions appears utterly difficult or rather impossible. If this is what Penrose actually has done, he should have the whole Prize. Or why did he not get the Prize long ago with Stephen Hawking? 

PS1 Here Penrose connects to the 2nd law of thermodynamics with a black hole swallowing entropy!

PS2 Penrose claims emphatically that quantum mechanics is inconsistent, or more precisely not a theory about physics! What then about relativity! 

PS3 Penrose area or science is presented to be mathematical physics. So what is this? It is not mathematics, because it lacks the rigour and strict logic of mathematics. It is not physics, because it is focussed on mathematical models per se and experimental verification is secondary. So what is it then? 

PS4 More aspects on the proof by Penrose:

• His 1965 paper actually used a proof by contradiction. He derived five properties needed in order for the system to avoid forming a singularity. Then he showed they are mutually inconsistent—a proof by contradiction.

Now proofs by contradiction have been regarded with suspicion through the history of mathematics, with constructive mathematics as mathematics without proofs by contradiction.  Many proofs by contradiction are proofs of existence of something, such as existence of a solution to some equation. Such proofs does not exhibit a solution, only says that there is one, while a constructive proof constructs/exhibits the solution, and thus contains much more concrete information.

To give an example consider this proof of the existence of God, which once was very convincing to people with big brains: God is by definition almigty. A property of almigtiness is existence, since almigtiness without existence would be a contradiction. Hence God exists. End of proof. 

Do you buy it? So to be sure that black holes do exist, it would be more convincing with a constructive proof showing how they are formed. 

Compare a Nobel Prize to a proof by contradiction of: there are planetary systems, with a Prize to the discovery of: how planetary systems are formed.  

 

Nobel Prize in Physics for Discovery of Singularity without Physics

The mathematician Roger Penrose receives half of the 2020 Nobel Prize in Physics

• for the discovery that black hole formation is a robust prediction of the general theory of relativity

with the following further motivation:

• In January 1965, ten years after Einstein’s death, Roger Penrose proved that black holes really can form and described them in detail; at their heart, black holes hide a singularity in which all the known laws of nature cease. His ground-breaking article is still regarded as the most important contribution to the general theory of relativity since Einstein.

So Penrose receives the Prize/2 for proving: in a singularity all the known laws of nature cease. Here is an illuminating illustration from the deep scientific text underlying the motivation (watch and get enlightened):

To give perspective on this perplexing message, recall that Einstein was awarded the 1921 Nobel Prize in Physics with the explicit mention that it was not for his theories of relativity. The 1921 Nobel Committee did not consider Einstein's relativity theories to be physics, only mathematics closed into its own box. It is logical that the 2020 Nobel Committee gives the Prize/2 to a mathematician. 

Maybe it can be seen as a little compensation for the lack of Nobel Prize in Mathematics. The 2020 Nobel Committee concludes the theoretical presentation with the following reservation in the spirit of the 1921 Committee: 

• The extent to which the structure of a black hole surrounded by an event horizon actually match the predictions of general relativity is still an open question. Nature may still have surprises in store.

For 100 years the Nobel Physics Committee stubbornly resisted giving the Prize to Einstein's Theories of Relativity, despite its proclaimed fundamental role in modern physics. What made the Committee change mind? Was it this (criticised) picture:

Below you find the Nobel Diploma to Einstein with the explicit mention on the first page that the Prize is 

• independent of the value, after eventual confirmation, which can be given to relativity/gravitation theory. 

It is certainly unique in the history of the Nobel Prize to  explicitly state for which landmark contribution to science the Prize is not given!

For more juice, see Ant Elzinga's interesting book Einstein's Nobel Prize, A Glimpse Behind Closed Doors.

Einstein's Biggest Mistake

Einstein described the introduction of the cosmological constant in his field equations of general theory of relativity as his Biggest Mistake. 

But that was a mistake: His truly Biggest Mistake was the introduction of his special theory of relativity in 1905, which preceded the general theory by ten years.

Special relativity concerns the representations of "events" in two space-time coordinate systems (x,t) and (x',t'), where an "event" (of some nature) is recorded by specific coordinates such as (0,0). 

The key example used by Einstein was the setting of train traveling along an embankment with a velocity $v<1$, with the x-axis fixed to the embankment and the x'-axis fixed to the train. The x-axis and x'-axis use the same spatial scale set by identical meter sticks. Events would then be recorded by an observer X on the embankment using the x-axis and a clock showing t-time, and by an observer X' in the train using an x'-axis  fixed to the train together with an identical clock traveling with the train showing t'-time. 

The windows of the train are supposed to be shut, so that X cannot look inside the train, and X' cannot look outside the train. X and X' thus use identical space-axes and identical clocks.  X and X' agree on a common speed of light = 1 as measured in their respective system with identical meter sticks and clocks. 

For a third outside observer Y (Galileo) the full situation would be clear and simple: The connection between the coordinates would be x' = x - vt and t' = t, expressing that the train with its x'-axis translates with respect to the x-axis of the embankment with velocity v and that identical clocks are being used. 

To understand that X and X' can agree on the speed of light, although the x'-axis moves with respect to the x'axis, compare with the speed of sound. X and X' will agree on the speed of sound, because the air inside the train travels along with train and so X' will measure the same speed of sound as X measures in the still air fixed to the embankment. Moreover, if X' opens a window and measures the speed of sound in the still air of the bank, X' would get a value modified by the train velocity, and X' would also experience a Doppler effect of sound propagation in the still air of the bank. All in accordance with the Galilean transformation x' = x -vt, and t' = t. 

But Einstein was not happy with a Galilean transformation between coordinates, because a medium of propagation of light (like air for propagation of sound), a unique "luminiferous aether ", could not be identified (in the Michelson-Morley experiment).  Einstein then came up with the radical idea of not connecting any form of medium to light propagation and then replaced the Galilean transformation, based on an embankment-medium and a different train-medium, by the Lorentz transformation supposedly being free of any medium: 

• $x^\prime =\gamma (x-vt)$, $t^\prime = \gamma (t-vx)$, $\gamma =\frac{1}{\sqrt(1-v^2)}$,
• $x =\gamma (x^\prime+vt^\prime )$, $t = \gamma (t^\prime+vx^\prime )$.

where $\vert v\vert <1$. The Lorentz transformation has the property that a $x = t$ is transformed into $x^\prime  = t^\prime$,  which is viewed to express the same speed of light = 1. Lorentz had introduced his transformation well before Einstein took it up, but Lorentz had been careful to note that his transformation was not to be interpreted as a transformation between physical coordinates. In particular, if t was physical time measured by a physical clock, then t' did not represent physical time measured by a physical clock. 

But the young eager brave Einstein took the bold step of proclaiming, against Lorentz, that both t-time and t'-time represent physical time measured by identical clocks. In particular with x=t recording a light signal in the (x,t)-system, t' would be connected to t by 

• $t' = \sqrt{\frac{1-v}{1+v}}t$

with thus $t' < t$ expressing that a moving X'-clock would be running slow compared to a still X-clock. This weird effect was termed time dilation: A moving clock would thus run slow compared to a stationary clock. Which clock was stationary and which was moving, was left in the air as an expression of Einstein's genius.

Having thrown out any form of medium for light propagation, Einstein also had to get rid of the observer Y capable of grasping the full physics of light propagation in the combined train-embankment system, and Einstein was then left with an observer X on the embankment unable to look into the train and an observer X' in the train unable to look out, as presented above.

X thus cannot make any observations in the (x',t')-system, nor can X' make any observations in the (x,t)-system. If either of them could, then the very reason to use two coordinate systems would disappear: If X' can make observations in the (x,t)-system, then X' does not need the (x',t')-system and vice versa. 

Einstein was thus left with two observers X and X' locked into two different systems unable to connect in physical terms. This is where Einstein takes a leap and boldly claims that observations are to be connected by the Lorentz transformation. Einstein thus tells X' inside the train that a light signal launched by X along the embankment, which X' cannot see, has to be represented in the (x',t')-system through the Lorentz transformation coming with space contraction and time dilation.

More precisely, to support his argument Einstein made the thought experiment that both X and X' launch light signals in their respective (x,t) and (x',t')-systems at (x,t) = (0,0)  and (x',t') = (0,0), which are then recorded as x=t and x'=t' respectively, which Einstein (but not Lorentz!) tells are connected by the Lorentz transformation. 

X' (in the train) thus observes a light signal x' = t' travelling inside the train, and X' is told by Einstein that this is the same light signal as that launched by X following x = t along the embankment with x = t. X'  is thus told by Einstein that his t'-clock is running slow as compared to the t-clock as an expression of weird  time dilation of his moving clock.  

But X' cannot observe the light signal traveling in the (x,t)-system (windows are shut), and Einstein's Biggest Mistake thus boils down to identifying the light signal launched by X in the (x,t)-system at (x,t) = (0,0), with the light signal launched by X' in the (x',t')-system at (x',t') = (0,0). Doing so and connecting the coordinates of the light signals by the Lorentz transformation, Einstein discovers the new physics of space contraction and time dilation.

But it is not correct to identify the two light signals, by the same reason that it would not be correct to identify two sound signals, one launched in the still air of the embankment and one in the moving air inside the train. Einstein's argument for identification is weak: Since there is no medium for light propagation, one can assume that the two signals are the same. More precisely, Einstein's identification lacks any form of rationale. 

Unfortunately modern physics is built on the special theory of relativity formed from Einstein's Biggest Mistake.

Luckily, there is a different rational physics of relativity of light propagation in the spirit of Galilean physics as Many-Minds Relativity.  Take a look and get enlightened (by light signals)! A short course is 

1. Special Relativity: There is no aether for light propagation.
2. Many-Minds Relativity: There are many aethers for light propagation, one for each choice of space coordinate system.