How to Measure Temperature

Measuring temperature accurately is a delicate procedure.

This is a comment to the discussion in recent posts of the proclaimed perfect blackbody spectrum of Cosmic Microwave Background CMB radiation with temperature 2.725 K.  

You can measure your body temperature by body contact with a quicksilver thermometer or at distance by an infrared thermometer. Both work on a principle of thermal equilibrium between source and thermometer sensor as a stable state over time. Your body is assigned the temperature recorded by the thermometer. 

Temperature can be seen as a measure of energy in the form of heat energy or vibrational energy of a vibrating system like an atomic lattice as the generator of radiation as radiative heat transfer.

Computational Blackbody Radiation offers a new analysis of radiative heat transfer using classical wave mechanics as a deterministic form of Planck's analysis based on statistics of quanta. The basic element of the analysis is a radiation spectrum from a vibrating atomic lattice: 

• $E(\nu ,T)=\gamma T\nu^2$ for $\nu \le \frac{T}{h}$        (1a)
• $E(\nu ,T)= 0$ for $\nu >\frac{T}{h}$                               (1b)

where $\nu$ is frequency on an absolute time scale, $T$ is temperature on a lattice specific energy scale, $\gamma$ and $h$ are lattice specific parameters and $\frac{T}{h}$ is a corresponding high-frequency cut-off frequency setting a upper limit to frequencies being radiated. Here a common temperature $T$ for all frequencies expresses thermal equilibrium between frequencies. 

It is natural to define a blackbody BB to have radiation spectrum of the form (1) with maximal $\gamma$ and high-frequency cut-off and to use this as a universal thermometer measuring the temperature of different bodies by thermal equilibrium. 

Consider then a vibrating atomic lattice A with spectrum according (1)-(2) with different parameters $\bar\gamma <\gamma$ and $\bar h >h$ and different temperature scale $\bar T$ to be in equilibrium with the universal thermometer. The radiation law (1) then implies assuming that A is perfectly reflecting for frequencies above its own cut-off:

• $\bar\gamma \bar T = \gamma T$                                         (2)

to serve as the connection between the temperature scales of BB and A. This gives (1) a form of universality with a universal $\gamma$ reflecting the use of a BB as a universal thermometer.

In reality the abrupt cut-off after at radiation maximum is replaced by a gradual decrease to zero over some frequency range as a case-specific post-max part of the spectrum.  A further case-specific element is non-perfect reflectivity above cut-off. Thermal equilibrium according to (2) is thus an ideal case.  

In particular, different bodies at the same distance to the Sun can take on different temperatures in thermal equilibrium with the Sun. Here the high-frequency part of the spectrum comes in as well as the route from non-equilibrium to equilibrium. 

Why CMB can have a perfect blackbody spectrum is hidden in the intricacies of the sensing. It may well reflect man-made universality. 

Man-Made Universality of Blackbody Radiation 2

Man-made Universality of Shape

This is a clarification of the previous post on the perfect Planck blackbody spectrum of the Cosmic Microwave Background Radiation CMB as a 14 Billion years afterglow of Big Bang as the leading narrative of cosmology physics today. See also this recent post and this older illuminating post.

The Planck spectrum as the spectrum of an ideal blackbody, takes the form 

• $E(\nu ,T) =\gamma T\nu^2\times C(\nu ,T)$                                         (1)

where $E (\nu ,T)$ is radiation intensity depending on frequency $\nu $ and temperature $T$, $\gamma$ a universal constant, and $C(\nu ,T)$ is a universal high frequency cut-off function of the specific form 

• $C(\nu ,T)=\frac{x}{\exp(x)-1}$ with $x = \frac{\nu}{T}\times\alpha$       (2)

where $\alpha =\frac{h}{k}$ with $h$ Planck's constant and $k$ Boltzmann's constant as another universal constant, with the property that 

• $C(\nu ,T)\approx 1$ for $x<<1$ and $C(\nu ,T)\approx 0$ for $x>>1$.  

We see that radiation intensity proportional to $T$ increases quadratically with $\nu$ in accordance with deterministic wave mechanics, and reaches a maximum shortly before a cut-off scaling with $T$ in accordance with statistics of energy quanta, which kicked off an idea of atom physics as quantum mechanics also based on statistics.    

Computational Blackbody Radiation offers a different version of high frequency cut-off motivated by finite precision physics/computation instead of statistics of quanta opening to a deterministic form of atom physics as real quantum mechanics. The underlying physics model in both cases is that of an atomic lattice capable of generating a continuous spectrum of vibrational frequencies.

The basic assumptions behind a Planck spectrum as an ideal are:

1. Model: Atomic lattice.
2. Equilibrium: All frequencies take on the same temperature.
3. High-frequency universal cut-off: Statistics of energy quanta.  

Observation show that most real blackbody spectra substantially deviate from the Planck spectrum and so have their own signature reflecting specific atomic lattice, non-equilibrium and specific high frequency cut-off lower than the ideal. Graphite is just about the only substance showing a Planck spectrum. 

This was not welcome by physicists in search of universality, and so the idea was born of deciding the spectrum of a given material/body by putting it inside an empty box with graphite walls and measuring the resulting radiation peeping out from a little hole in the box, which not surprisingly showed to be a graphite Planck blackbody spectrum. 

Universality of radiation was then established in the same way as universality of shape can be attained by cutting everything into cubical shape as was done by the brave men cutting paving stone out of the granite rocks of the West Coast of Sweden, which is nothing but man-made universality.  

The line spectrum of a gas is even further away from a blackbody spectrum. The idea of CMB as an afterglow of a young Universe gas cloud with a perfect Planck blackbody as measured by the FIRAS instrument on the COBE satellite, serves as a corner stone of current Big Bang + Inflation cosmology. 

It is not far-fetched to suspect that also the COBE spectrum is man-made, and then also Big Bang + Inflation.

Can Cosmic Microwave Background Radiation be Measured, Really?

The Cosmic Microwave Background radiation CMB is supposed to be a 14 billion year after-glow with perfect Planck blackbody spectrum at temperature $T=2.725$ Kelvin K of a Universe at $T=3000$ K dating back to 380.000 years after Big Bang. The apparent 1000-fold temperature drop from 3000 to 3 K is supposed to be the results of an expansion and not cooling.  

To get an idea of the magnitude of CMB let us recall that a Planck spectrum at temperature $T$ stretches over frequencies $\nu\sim T$ and  reaches maximum radiation intensity $E\sim T^3$ near the end with a high frequency cut-off over an interval $\frac{\nu}{T}\sim 1$ (notice exponential scale):

 

The $10^3$-fold temperature drop thus corresponds to a $10^9$ decrease of maximum intensity and $10^3$ decrease in spectrum width. Intensity over width decreases with a factor $10^6$ as a measure of precision in peak frequency. 

We understand that to draw conclusions concerning a 3000 K spectrum from a measured 3 K spectrum requires a very precision on the level of microKelvin or 0.000001 K. Is this really possible? Is it possible to reach the precision 2.725 K from intensity maximum? 

Why is modern physics focussed on measuring quantities which cannot be measured, like ghosts?

CMB was first detected as noise maybe from birds visiting antennas, but the noise persisted even after antennas were cleaned and then the conclusion was drawn that CMB must be left-over from Big Bang 14 billion years ago and not from any birds of today.  Big Bang is physics, while birds is ornithology. 

The Ultra-Violet Catastrophe vs 2nd Law of Thermodynamics

Classical physics peaked in the late 19th century with Maxwell's equations aiming to describe all of electromagnetics as a form of continuum wave mechanics, but crumbled when confronted with the Ultra-Violet Catastrophe UVC of heat radiation from a body of temperature $T$ scaling like $T\nu^2$ with frequency $\nu$ threatening to turn everything into flames without an upper bound for frequencies, because wave mechanics did not seem to offer any escape from UVC.  

Planck took on role of saving physics from looming catastrophe, but could not find a resolution within deterministic wave mechanics and so finally gave up and resorted to statistical mechanics with high frequencies less likely in the spirit of Boltzmann's thermodynamics and 2nd Law with order less likely than disorder. 

There is thus a close connection between UVC and 2nd Law. Boltzmann would say that the reason we do not experience UVC is that high frequencies are not likely, but the physics of why is missing. Explaining that UVC is not likely would no explain why there is not any observation UVC whatsoever. 

I have followed a different route replacing statistics by finite precision physics for UVC (and similarly for 2nd Law), where high frequencies with short wave length cannot be radiated because finite precision sets a limit on the frequencies an atomic lattice can carry as coordinated synchronised motion. In this setting UVC can never occur.

A basic mission for a 2nd Law is thus to prevent UVC. This gives 2nd Law deeper meaning as a necessary mechanism preventing too fine structures/high frequencies to appear and so cause havoc. 2nd Law is thus not a failure to maintain order over time, but a necessary mechanism to avoid catastrophe from too much order. 

Similarly, viscosity and friction appear as necessary mechanisms destroying finite structure/order in order to let the World to continue, and so not only as defects of an ideal physics without viscosity and friction. This is the role of turbulence as described in Computational Turbulent Incompressible Flow and Computational Thermodynamics.

We can compare with the role of interest rate in an economy with zero interest rate of an ideal economy leading to catastrophe over time. If there is no cost of getting access to capital, any crazy mega project could get funding and catastrophe would follow. This was the idea 2008-2023 preceding the collapse predicted to 2025. Too little friction makes the wheels turn too fast. Too much idealism leads to ruin.

The Secret of Radiative Heat Transfer vs CMB and Big Bang

A main challenge to physicists at the turn to modernity 1900 was to explain radiative heat transfer as the process of emission, transfer and absorption of heat energy by electromagnetic waves described by Maxwell's equations. The challenge was to explain why real physics avoids an ultra-violet catastrophe with radiation intensity going to infinity with increasing frequency beyond the visible spectrum. 

More precisely, the challenge was to uncover the physics of a blackbody spectrum with radiation intensity scaling with $T\nu^2$ with $T$ temperature and frequency $\nu\le\nu_{max}$ with $\nu_{max}$ a cut-off frequency scaling with $T$, and intensity quickly falling to zero above cut-off. 

Planck as leading physicist of the German Empire took on the challenge and after much struggle came up with an explanation based on statistics of energy taking the above form as Planck's Law, which has served into our time as a cover up a failure to explain a basic phenomenon in physical terms. 

Computational Blackbody Radiation offers an explanation in terms of finite precision physics setting a cut-off (scaling with temperature) on the frequency of emission from coordinated oscillations of an atomic lattice, with uncoordinated atomic motion stored as heat energy.

In this analysis heat is transferred from a body of higher temperature  to a body of lower temperature through a resonance phenomenon analogous to the resonance between two tuning forks. The essence can be described in terms of a  forced acoustically weakly damped harmonic oscillator:

• $\dot v(t)+\nu^2u(t)+\gamma v(t)=f(t)=sin(\bar\nu t)$ for $t>0$                    (1)

where $u(t)$ is displacement at time $t$, $v(t)=\dot u(t)$ is velocity, the dot represents derivative with respect to time $t$, $\nu$ is the frequency of the harmonic oscillator and $\bar\nu\approx\nu$ that of the forcing. For radiation the damping term takes the form $\gamma\ddot v(t)$. 

Mathematical analysis shows assuming small damping with $\gamma << 1$ and near resonance with $\nu\approx\bar\nu$ and integration over a period:

• $Output = \gamma \int v^2(t)dt \approx \int f^2(t)dt = Input$         (2)
• Velocity $v(t)$ out-of-phase with $f(t)$.                                                                (3)

Even if it looks innocent, (2) represents the essence of Planck's Law with (3) expressing basic physics: Out-of-phase means that the interacting between forcing and oscillator corresponds to a "pumping motion" with the forcing balanced mainly by the harmonic oscillator itself and not the damping. In the acoustic case $T=\int v^2(t)dt$ and thus $Output =\gamma T$, which in the case of radiation takes the form $Output = \gamma T\nu^2$ or Planck's Law. 

Sum up:

• Radiative balance between two bodies of equal temperature is expressed by (2).
• Heating of a body B1 with lower temperature from body B2 of higher temperature from frequencies above cut-off for B1.  
• High frequency cut-off effect of finite precision physics and not statistics.
• Blackbody spectrum is continuous (all frequencies) and requires atomic lattice. 
• A gas ha a line spectrum with selected frequencies, which is not a blackbody spectrum.
• Cosmic Microwave Background radiation as a perfect blackbody spectrum of an after-glow of Big Bang without atomic lattice appears as very speculative, with Big Bang itself as even more speculative beyond experimental confirmation.  

Does a Photon have Temperature?

The idea about the Cosmic Microwave Background CMB radiation is conveyed to the public by authoritative sources as follows starting at the creation of the Universe with a Big Bang:

• After about 380,000 years when the Universe had cooled to around 3000 Kelvin,  photons were able to move unhindered through the Universe: it became transparent.
• Over the intervening 14 billion years, the Universe has expanded and cooled greatly. Due to the expansion of space, the wavelengths of the photons have grown (they have been ‘redshifted’) to roughly 1 millimetre and thus their effective temperature has decreased to just 2.7 Kelvin. 
• These photons fill the Universe today (there are roughly 400 in every cubic centimetre of space) and create a background glow that can be detected by far-infrared and radio telescopes.

We meet the idea that photons are moving through space like some form of particles with effective temperature of 2.7 K filling the Universe as an after-glow of Big Bang. 

But the concept of photon lacks real physics. Light does not consist of a stream of light particles named photons, but is an electromagnetic wave phenomena and as such can have a frequency and an amplitude/intensity. An emitter of light like the Sun has a temperature, while the light emitted is characterised by its spectrum as intensity vs frequency. A spectrum can give information about the temperature of the emitter with the Planck spectrum the spectrum of an ideal blackbody at a certain temperature with in particular a high-frequency cut-off scaling linearly with temperature. 

Emitted light can be recieved by an antenna through resonance recording the frequency. It is also possible to record the temperature of an emitter by connecting the antenna to a form of radiation thermometer reading temperature from radiative equilibrium, in the same way as a common thermometer reads the temperature of a source by direct contact/equilibrium.  

But is more difficult to read a spectrum since properties of emissivity, transmissivity and absorptivity as well as view angles enter. In the absence of information a Planck spectrum is often assumed, but most emitters do not have blackbody spectra.

A Big Bang emitter at 3000 K is thus postulated with an after-glow received as a blackbody spectrum of 3 K with frequency reduced and wave length increased by a factor of 1000 into far-infrared. 

What is effectively measured is a combination of temperature and intensity, which shows up as a perfect blackbody spectrum. The message is that this is an after-glow of Big Bang, thus giving evidence to Big Bang: If there is an after-glow there must have been some glow to start with = Big Bang. More precisely, it is variations letting the antenna sweep the sky, which are measured and have to be given a physical meaning as some variability of the Early Universe. 

The basic idea is thus that photons have been traveling through empty space for 14 billion years under a stretching of a factor 1000 but no other influence, and that collecting these photons gives a picture of the Early Universe. This appears as a lofty  speculation cleverly designed as to prevent inspection because both theory and instrumentation are hidden in mist. Here is the main picture from The Music of the Big Bang by Amedo Balbi: 

The source is thus gone since 14 billion years, while the after-glow still surrounds us and can be measured. This is mind boggling. 

Let us compare with the picture presented as Computational Blackbody Radiation, where emitter and receiver establish contact by resonance of electromagnetic waves and so take on the same temperature by reaching radiative equilibrium, in the same way as two distant tune forks can find an equilibrium.

What about the time delay between emitter and receiver from finite speed of light? If a light source is switched on, it will take some time before it reaches a receiver. Is it the same when a light source is switched off? Do you feel being warmed even a while after the fire is dead? What about a solar eclipse? Does it take 8 minutes before we feel the cold? 

In any case, the connection between Big Bang which is gone since 14 billion years and a proclaimed after-glow, which we can enjoy today from the presence of about 400 photons in every cubic centimetre of space at 3 K, appears as science fiction to me at least. 

Radiation as electromagnetic waves needs a source to sustain over time. If the Big Bang source to CMB disappeared 14 billion years ago, the electromagnetic waves have so to speak have a life of their own over very long time, like a tsunami wave sweeping the Pacific long after the earth quake source has disappeared. Here the ocean acts as a physical medium carrying the energy, while a corresponding medium for electromagnetic waves as an aether has no physical presence. The energy is thus carried by the source some of which is transmitted to the receiver in resonance. 

Modern Physics vs Homeopathy

Modern physics appears as a form of homeopathy in reverse. A main idea of homeopathy is to obtain a major health effect from a very very diluted form of some substance, the smaller the better. The characteristic of modern physics, as rather the opposite, is identification of a very small effect from a very very large cause as in the following key examples  with increasing presence in later years (with year of Nobel Prize in Physics):

• Theoretical and experimental discovery of very small deviation from Newton's mechanics in Einstein's mechanics. (no Prize)
• Theoretical discovery of the Pauli Exclusion Principle impossible to verify experimentally. (1945)
• Theoretical discovery of statistical interpretation of wave function impossible to verify experimentally. (1954)
• Experimental discovery of Microwave Background. (1978)
• Experimental discovery of very small fluctuation in temperature of Cosmic Microwave Background Radiation from Big Bang. (2006)
• Theoretical discovery of broken symmetry predicting quarks impossible to verify experimentally. (2008)
• Experimental discovery of accelerating expansion of the Universe from very weak data. (2011)
• Experimental discovery of Higg's particle as origin of mass from very weak data. (2013)
• Experimental discovery of very weak gravitational waves from collision of supernovae. (2017)
• Theoretical discovery that black hole is a prediction of general relativity. (2020)
• Theoretical discovery that global warming is a prediction of very little atmospheric CO2. (2021)
• Theoretical discovery of string theory on very very small scales impossible to verify experimentally. (no Prize)

It may seem that all notable effects have already been discovered and so only very very small remain to be discovered. The difficulty of connecting a very small effect to a very large cause (or vice versa) is that a very precise theory is needed in order to build a very precise instrument for experimental verification. Without theory experiment has no objective. Finding a needle in a haystack may be simpler. In addition to experimental discoveries of some vanishingly small effect, we also see Prizes to discoveries of theories beyond experimental verification because effects are so small.  

When the Large Hadron Collider shows to be too small to find anything new of significance and public money for an even larger Super Collider cannot be harvested, physicists turn to use the whole Universe as test bench to find ever smaller effects. There are many things yet to be discovered on scales allowing detection, but this draws little interest from leading physicists focussed on what is infinitely large or infinitesimally small.  

We may compare with the evaluation by Napoleon of the work in his administration of the mathematician Laplace as expert on Infinitesimal Calculus: 

• He wanted to bring the spirit of infinitesimals into administration.

Cosmic Microwave Background Radiation vs Big Bang?

This is a continuation of a previous post on the same topic. The European Space Agency ESA sends this message to the people of Europe and the World:

• The Cosmic Microwave Background (CMB) is the cooled remnant of the first light that could ever travel freely throughout the Universe.
• Scientists consider it as an echo or 'shockwave' of the Big Bang. Over time, this primeval light has cooled and weakened considerably; nowadays we detect it in the microwave domain.

More precisely, CMB is reported to be measured by the FIRAS Far Infrared Absolute Spectrophotometer (FIRAS) instrument on the COBE satellite as a very small temperature variation (18 $\mu K$) over a uniform background of a perfect blackbody spectrum at 2.725 $K$. The main difficulty is to isolate a very weak signal from very far away from more nearby signals including signals from the Earth atmosphere and oceans.  

To understand the technology of the measurement, which is not easy, we take a look at the FIRAS instrument to see what it contains:

 What we see is in particular the following:

• Sky Horn collecting input from the Sky.
• Xcal reference blackbody used for calibration of Sky Horn input.
• Ical reference blackbody for internal calibration.
• Ical is equipped with two germanium resistance thermometers (GRT).
• Xcal is monitored by three GRTs.
• FIRAS = Far Infrared Absolute Spectrophotometer.

The output of FIRAS consists of:

• A very small temperature variation of size 0.00002 K over a background of 2.725 K.
• The measured background spectrum is a perfect Planck blackbody spectrum. 

CMB spectrum as perfect Planck blackbody spectrum. But low frequencies in the far infrared spectrum are missing! 

We see warning signs: 

• Very high precision is reported!
• Perfect Planck blackbody spectrum is reported. But far infrared is missing. 
• Calibration to nearly perfect real blackbodies is made. 
• Temperature of 3 K from very far reported.  
• Spectrum as radiative flux is reported (spectrophotometer).

More understanding comes from plotting the spectrum in terms of frequency:

We here see the COBE-FIRAS (blue) measures intensity at maximum around 200 GHz and a bit beyond for higher frequencies in the cut-off region, while the more essential part of the spectrum in the far infrared is missing. The intensity maximum around 200GHz according to Planck's law corresponds to a temperature of about 3 K, which however, since the essential part of the spectrum is missing, may as well correspond to much higher temperature at much lower emissivity.

In previous posts we have reminded that measurement of temperature is possible by establishing radiative equilibrium between source and instrument/thermometer, but it requires disturbances between source and instrument to be small, which poses a challenge to directly measuring temperature of CMB from very far. 

The alternative in this case is to report temperature from spectrum. But directly measuring radiative flux/spectrum can be even more challenging. Typically this is done (using bolometers and pyrometers) by measuring temperature, and then computing radiative flux/spectrum using Planck's law under assumptions hard to verify. This makes assessing CMB to a very daunting task from a mix of measurement and computation of temperature and radiative flux.

The scenario is thus:

• If a correct full spectrum is measured, a temperature can be determined from the frequency of maximal intensity. 
• If only temperature is given, determining spectrum as radiative flux intensity, requires post processing. 
• A measured/computed temperature of 3K attributed to a very far away source may be misleading.
• Robitaille suggesting that the true origin of the the 3 K CMB is the oceans of the Earth at 300 K.  

To sum up, we have on the table: 

1. Very speculative Big Bang BB.
2. CMB with questionable credibility, maybe noise from Earth Ocean,  

The argument by mainstream physicists/cosmologists is now that since the main role of CMB is to serve as main evidence of Big Bang, and CMB shows to serve this role in such an excellent way, it gives credibility to CMB by being connected to something very big. BB thus supports CMB, which gives support to BB. 

One possibility is then that both BB and CMB are real phenomena The other possibility is that both are free speculations by scientists in search of a mission. What is your impression? 

PS Has COBE-FIRAS detected the same thing as WMAP and PLANCK further away from the Earth:

Which picture is most credible? The more details, the more credible? What happens with small details over time according to the 2nd Law? 

Computing with Real Numbers

This is a continuation of the previous post How to avoid collapse of modern mathematics.

Let me see if the constructive computational approach to mathematics adopted in the BodySoul program can meet the criticism expressed by Norman Wildberger as concerns the foundations of the large areas of mathematics relying on the concept of real number.  In particular Wildberger asks about the elementary process of adding two real numbers such as $\sqrt{2}$ and $\pi$: 

• $\sqrt{2}+ \pi = ?$ 

Let us then use the least cryptic definition of a real number as an infinite decimal expansion. But asking for the infinite decimal expansion of $\sqrt{2}$ is asking too much, and so we have to limit the specification to a finite number of decimals, and the same with $\pi$. We can then add these numbers using well specified rules for computing with rational numbers, and so arrive at a finite decimal expansion as an approximation of $\sqrt{2}+ \pi$. We can choose the number of decimals to meet a given precision requirement. Fair enough. 

But how do we know the decimal expansions of $\sqrt{2}$ and $\pi$?. Before the computer they would have to be picked up from a printed precomputed mathematical table, but only up to finitely many decimals and the table would swell beyond all limits by asking for more and more decimals. Today with the computer, you can press a button and let $\sqrt{2}$ be computed from scratch using Newton's method, but even if this algorithm is very efficient, the required work/time would increase beyond limit by asking for more and more decimals. 

The computer would compute the sum $\sqrt{2}+ \pi$ in an iterative computational process involving:

1. Compute $\sqrt{2}$ with say $5$ decimals.
2. Compute $\pi$ with say $5$ decimals.
3. Add these decimal expansions using the addition algorithm for finite decimal expansions.
4. Check if a desired precision is met, and if not go back to 1. and increase the number of decimals.  

This would reduce the foundation of mathematics to computational processes, and this is the approach of BodySoul: All mathematical objects are constructed by specified finitary computational processes as finite precision solutions to specified equations. 

For example, the value of the exponential function $\exp(t)$ for any value $t>0$ is computed by solving the differential equation $x^\prime (s)=x(s)$ for $0<s\le t$ with $x(0)=1$ by time stepping, where $x^\prime $ is the derivative of $x$, and setting $\exp(t)=x(t)$. No values of $\exp(t)$ are stored. New computation from scratch for each value of $t$. This is the only way to avoid storing real numbers as infinite decimal expansions, which is impossible in a finite Universe. 

Is Wildberger happy with such a response to his criticism. And what about you?

In any case, pure mathematicians will not welcome a foundation based on non-pure computational mathematics, even if it would solve unresolved foundational questions concerning real numbers and elementary functions of real numbers as solutions to differential equations. 

There was a tough fight at the turn to to modernity in the beginning of the 20th century concerning the foundations of mathematics between logicism (Russell), formalism (Hilbert) and constructivism/intuitionism (Brouwer), which was won by Hilbert in the 1930s thus setting the scene for 21st century mathematics. But with the computer, constructivism is now taking over by offering a concrete foundation without lofty speculation of infinities.  

A formalist can introduce $\mathcal{R}$ as the set of equivalence classes of all Cauchy sequences of rational numbers thus as a set defined by a certain property. Russell showed the danger of defining sets this lofty way by his famous example of a set defined by the property of not containing itself leading to a contradiction. Gödel turned Russell's example into more precise form, which should have killed both the logicist and formalist school, but did not since the reaction was to kick out constructionists compatible with Gödel from mathematics departments to form separate departments of computer science. Mathematics departments/education is still controlled by formalists, which means that Wildberger's criticism is not welcome.    

Is Cosmic Microwave Background Radiation Measurable?

Temperature fluctuations of CMB measured by COBE satellite. 

Pierre-Marie Robitaille leading the development of the 8 Tesla Ultra High Field human MRI (Magnetic Resonance Imaging) scanner, used his deep knowledge of electromagnetic resonance to question the measurement of the Cosmic Microwave Background Radiation (CMB) by NASA's COBE satellite, awarded the Nobel Prize in 2008. 

This was not well received by the physics community and Robitaille was effectively cancelled academically (as far as I understand), but his very informative youtube channel Sky Scholar (with 50k subscribers and 145 videos) has survived. Take a look and compare with previous post on Man Made Universality of Black Body Radiation.

CMB is supposed to be the "cooled remnant of the first light that could ever travel freely throughout the Universe" at the very low temperature of 2.726 K above ultimate 0 K. Very cold indeed. More precisely, it is claimed that measured CMB spectrum is very close to a blackbody spectrum at 2.726 K. 

In previous posts I have posed the question if the spectrometer involved in measuring CMB is effectively measuring temperature or radiative flux, with the answer that temperature can be measured at distance by radiative equilibrium in the same way a thermometer in contact measures temperature and not heat flux by establishing radiative equilibrium. This is supported by the fact that it is a measured temperature of 2.726 K, which is the main characteristic of the postulated CMB, not its unknown radiative heat emission as a (small) possible contribution to global warming. Recalling previous posts and Robitaille, we know that the blackbody spectrum is a fiction only met by graphite and so one may ask why CMB could behave the same. 

In the view presented on Computational Blackbody Radiation the temperature measurement by NASA's COBE satellite as main evidence of the existence of CMB, is based on resonance between apparatus and cosmic background, which has to be singled out from all other resonances. Robitaille here presents the Oceans of the Earth as a possible source overwhelming CMB, thus questioning the existence of CMB.  

When your brain registers a sound arising from resonance between a sound source and eardrum, the direction to the source can be decided because you have two ears, but the distance to the source and so the origin of the sound is more difficult to determine in the presence of other possibly stronger sources.  Robitaille questions the possibility to single out CMB from the radiation from the Oceans.  Do you?