Math is True but Words can Lie

Written by Joseph E. Postma

While mathematics is a Formal language, English and any other verbal language are Natural spoken languages.  And with human languages, the inevitable result is that you can lie with them.  Because mathematics can be complicated and it is readily apparent that even people with PhD’s in science have a hard time understanding it, it is therefore possible to present a totally valid mathematical equation and at the same time totally misrepresent what the equation means.  This is, of course, the purview of sophistry and those who produce it.
equation tattoo
 
What I will do here is give you some simple math, and the correct words and correct descriptions to understand it, and then contrast that to some mental garbage that has instead been presented in order to lie about what the math actually means from some examples that I’ve been personally witness to.
 
Case Point #1
 
Let us look at the equation for radiant heat flow between a hot object and a cold object.  In the equation below, the scenario could be for two walls facing each other which have unit emissivities and absorptivities, so that these factors, and the areas, can all be cancelled out of the equation.  The equation is thus:
 
Q =  σ[ (Thot)4 – (Tcold)4 ]                                                                                                         Eq.{1}
 
and it couldn’t be any more simple.  It simply says that “Q”, which is the rate of heat transfer between a hot object and cold object, in Joules per second per square meter, is equal to a constant “σ” (sigma) times the difference of the fourth powers of the temperatures of the two objects.  This makes sense: the greater the difference in temperature, the more heating power the hotter object will have on the cooler object because it will be that much warmer than the cooler object.
 
In Equation 1, Thot and Tcold are called independent parameters, meaning that they’re determined independently of the equation itself, by measurement, say.  On the other hand, Q, the heat transfer rate, is a dependent parameter because it obviously depends on the values on the right hand side of the equation.  For example, if you increase Tcold (or decrease Thot), then you decrease Q because you made the difference between the hot temperature and cold temperature smaller.  Conversely, if you increase Thot (or decrease Tcold), then you increase Q because you made the difference between the hot temperature and cold temperature larger.  The equation is for telling you what the value of Q is given two temperatures, and so Q is not a fixed independent parameter but is rather dependent upon the two temperatures.
 
Greenhouse effect believers who apparently do not understand physics, although they can do some simple math, have stated that if you fix Q in that equation, and then increase Tcold, then Thot has to increase “in order to keep Q constant”, and “therefore cold heats up hot”.  This claim is made because they have this faith belief system that cold things make hotter things hotter still, rather than um, you know, hot things making cold things hotter still….(lol).  The person (a sophist) even went out of their way to rearrange the equation so that Q was no longer a dependent parameter on the left hand side of the equation, in order to make it look like this:
 
Thot = 4√[Q/σ + (Tcold)4]                                                                                                          Eq.{2}
 
All this is, is a simple algebraic rearrangement of Equation 1; doing such a thing does not change what the actual original physical equation represents in the first place.  The only way this simple algebraic rearrangement makes sense is if you were giving a problem to a student, in which you knew the temperature of the cold object and you also knew the current rate of heat transfer between the hot object and cold object, and were thus asked to determine the temperature of the hot object.  Problems like this are done simply for the training of mathematical competency and relating it to theoretical physical problems; the Q parameter in Equation 2 still depends on the difference between Thot and Tcold and can not in any way be independently fixed.
 
If you understood Equation 1, then it is clear that is impossible to “hold Q constant” if you increase Tcold.  To say that you want to hold Q constant in Equation 2, actually makes Q an independent source of input energy and heat that no longer has any relation whatsoever to the difference between Thot and Tcold and the heat transfer equation, and so that is a completely different problem and set of physical principles you’re dealing with.  Pretending that you can hold Q constant in that equation, in order to further pretend that cold heats hot and thus there is a greenhouse effect, is pure sophistry – albeit advanced sophistry.  It is outright lying with (or should I say, about) mathematics, in no uncertain terms.
 
How to actually do it
 
If you want a physical equation that denotes temperature as function dependent on external independent parameters, such as an independent fixed heat source “Q”, then you have to go through the development for such a thing as I showed in last year’s paper where we proved that there is no GHE in operation in the atmosphere.  I’ll quickly show this here.
 
Tobj = q / (m*Cp)                                                                                                                      Eq. {3}
 
where “q” (different from big Q; but see ahead) is just the internally held total thermal energy content of an object of mass m, with thermal capacity “Cp“, at temperature “Tobj“.
 
To know how the temperature changes as a function of a change in the internal energy, we take the differential with respect to time:
 
dTobj/dt = 1/τ * dq/dt                                                                                                            Eq. {4}
 
where τ = m*Cp for convenience.  Now, in terms of energy input and output and the first law of thermodynamics, the temperature will change when the time-derivative of q, dq/dt, which is the total rate at which energy is entering or leaving the system, is non-zero.  To follow the unit convention above from Equations 1 & 2, where big “Q” is a rate of heat transfer, then dq/dt = Q.  Q now represents the sum of independent and dependent energy inputs and outputs, and so can actually be composed of multiple terms – two terms if there is an input and output.
 
In terms of radiation, the energy output from the surface of the object is σ(Tobj)4, and so that is the output term of Q which is dependent on the object’s current temperature.  That leaves an input term for Q which can be an independent parameter which doesn’t depend on any other terms in the equation.  Changing the notation a little bit, Q can now just represent the independent input, while σ(Tobj)4 represents the dependent output.  So this gives us
 
dTobj/dt = 1/τ * (Qin – σ(Tobj)4)                                                                                            Eq. {5}
 
which is a non-linear differential equation.  The input term is positive because it will serve to increase the temperature, while the output term is negative because output provides cooling.  This is the only way in which you can speak of fixing an independent variable labelled “Q”; it works here because Q is a true independent variable which does not actually depend on the other terms in the equation.
 
To make this loook similar to our initial setup, if Tobj refers to a passive cold wall (Tcold), then Qin can refer to a hot wall with constant temperature, and then Qin = σ(Thot)4 leaving
 
dTcold/dt = σ/τ * ((Thot)4 – (Tcold)4)                                                                                      Eq. {6}
 
When the temperature of the cold wall increases, then all that happens is that the rate of increase of temperature of the cold wall decreases, because the difference in temperature between the hot wall and cold wall becomes smaller.  It is basically in this way that the condition of thermal equilibrium is achieved in nature.  And note that an increasing temperature of the cold wall does not affect the temperature of the independent hot wall!  Cold does not heat hot in real physics.
 
Now that we have a new equation, I should point out that it is obviously still possible for people to lie about what it means, misinterpret its use, and create greenhouse effect sophistry and obfuscation with it.  Of course, I know exactly how that would be done and what would be said, but I’ll save having to write about it for another article, when a sophist inevitably tries to do it.
 
Case Point #2
 
This development nicely brings us back to a recent example of someone trying to use mathematics to lie, and misrepresenting what the mathematical equations actually are, and what they say.
 
Equations 5 & 6 are differential equations and in a very simple mathematical sense they can be said to be “time dependent”, because they can be solved as functions of time.   However, and this is a big however, in terms of physics the equations are not actually physically time-dependent differential equations when the independent input term, Qin = σ(Thot)4, is a constant term, meaning that it has no time dependence.  If you’re actually trained as a physicist, then distinguishing which mathematical parameters actually have physical time dependence and which ones don’t is very important – it couldn’t be any more fundamental to physics.  A real physicist would not say that Equations 5 & 6 are time dependent, because none of the terms are explicit functions of time as used in the averaged-out constant cold sunshine models of the greenhouse effect.
 
And so this brings us to our good old friend of the most traditional greenhouse effect lie: constant cold sunshine on a flat static Earth!  The mathematical/physics lie here is in claiming that an averaged, constant power input, is the same thing close enough as a time-varying input, and that a differential equation using those constant terms is time-dependent, and of course that the Earth is flat or static (not rotating, no day & night), and that sunshine is too cold to be able to heat anything above -180C, etc.
 
Essentially, the greenhouse effect premises are just a big tangled plate of a spaghetti of lies.
 
For the rest of this, and other articles by Joe Postma please visit climateofsophistry.com

Comments (4)

  • Avatar

    DougCotton

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    The main mistake they make is to assume that the Second Law of Thermodynamics is not violated if some combination of components (or processes) has a “net” effect that simply ends up with a net transfer of hot to cold. This comes down to really just applying energy conservation (First Law) without the real restrictions of the Second Law of Thermodynamics.

    They can be pinned down on this by pointing out that the Second Law refers to a process in which thermodynamic equilibrium evolves spontaneously in an isolated [b]system[/b].

    But what is a system? Physics tells us that it is either a single component (process) or a set of [b]interdependent components.[/b]

    That puts the onus back on them to prove that the components are interdependent. For example, if the Sun warms some rock, which then warms some water, which then evaporates and subsequently releases latent heat, could all four components be interdependent? Hardly!

    A siphon represents two components which are interdependent. Water can flow up the shorter “component” provided that more water is simultaneously flowing down the longer component. Cut the hose at the top and you remove the interdependence – and water no longer flows uphill. Neither does back radiation transfer heat “uphill” and neither does water flow up a creek into a lake because it knows it has a bigger creek to flow down further on the other side of the mountain.

  • Avatar

    Joseph E Postma

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    Glad you have liked the article.

    Subbing to comments. Can comment at my blog too at the climateofsophistry link.

    Cheers!

  • Avatar

    J osullivan

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    Joe, a very instructive guideline for those who struggle with the math. Thanks!

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    Jacob

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    Great article, Joe. Very informative!

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