what is it and what to cook it with
Everyone knows what axioms are, but few people understand what they are.
The original formulation “an axiom is a proposition accepted as true without evidence” is interpreted as meaning that an axiom is something that is so unshakable and obvious truth that it does not require any evidence.
The problem with this interpretation is the word “is”, and here’s why.
Axioms at school
The most famous and best example of an axiom is the axiom of parallel lines.
The same one that we learn in school in the form “through a point not lying on a straight line, in the plane defined by this line and point, one can draw one and only one straight line that does not intersect the given straight line.”
We live in a world where this rule is followed, where it is used by science, technology and art, and we begin to believe that this is how it should be – that there is an objective reality and there is its indisputable reflection called an “axiom”.
Therefore, when we learn about non-Euclidean geometries, it makes a very big impression on us and makes us surprised. It turns out that there is some kind of mathematical theory in which the most obvious of truths is not recognized.
And all this surprise occurs due to the fact that we simply do not remember what we were told back then in geometry lessons.
And they told us that “accepted without evidence” does not mean “accepted as truth given from above”but strictly the opposite – “made by willful decision”.
Yes, you can say “one and only one line”, you can say “not one”, you can say “more than one” and by a volitional decision accept (that is, assign) this as truth in three different theories and logical ones.
But the fact that we can, by a volitional decision, accept any position as an axiom is not only the essence of the axioms, but one of the most important practical properties of the axioms.
And this was also in the school curriculum.
“Proof by contradiction” – we introduce an axiom that some statement is false and try to build an integral system that is consistent both internally and with what we consider reality.
“Neglect friction” – we introduce the axiom about the absence of friction, which is not only false within the framework of the theories studied in other academic subjects, but is something that we consider to be contrary to reality. And thanks to the fact that we did this, the calculation logic is greatly simplified.
Moreover, such treatment of axioms occurs not only in school lessons, but also in serious calculations.
For example, the basis of the often used “Equation of State of an Ideal Gas” is the axiom that gas is considered as a monolithic entity and does not consist of molecules with mass, volume and other material properties. Thanks to this, we have a simple and convenient equation.
And also, when calculating ventilation in residential buildings and industrial premises, air is considered not as a “gas”, but as an “incompressible liquid”. A position contradicting physical reality was introduced into the axiomatics of ventilation calculations and a convenient and practical mathematical apparatus was obtained.
That is, the axiom is not “the way it really is,” and not even “what looks like it actually is.” The axiom is “within the framework of this calculation/project/theory, we will proceed from this, and it doesn’t matter how it really is.”
But besides the triumph of voluntarism (and possibly opportunism), another important property of axioms follows from “accepted without proof.”
Euclid’s fifth postulate
Euclid’s fifth postulate is the same axiom about the number of lines that can be drawn through a point (also known as the “axiom of parallel lines”).
The fact is that the idea formulated in it is so obvious, so on the surface, that there is a feeling of its regularity. And if something is regular, then there is a temptation to break this pattern down into smaller parts and prove it.
And here again the problem of interpreting axioms as what actually exists arises. Because, in this case, the idea of “proving Euclid’s fifth postulate” takes on a mystical touch of “proving reality” and “knowing the truth.”
In fact, the topic of “proof of the axiom of parallel lines” is not about mysticism or objective reality. What then?
Well, firstly, for mathematicians it is a matter of sports passion, professional pride and the desire to place themselves in the pantheon of mathematicians of all times and peoples.
And secondly, it is about that very property of axioms “accepted without proof.”
Well, look, you have 5 axioms on which you built all the geometry. These are theorems, equations and dividing an angle using a compass, they are built on 5 axioms.
You make a conclusion from 2 axioms, another conclusion from 3 other axioms, then you make another one from these conclusions, then you add another pinch of axioms and another conclusion. And so, step by step, build the entire geometry, using axioms as building blocks. Somewhere the bricks are used themselves, and somewhere in the form of an already built wall with a window and a door to the loggia.
And what happens if you manage to prove the fifth axiom?
That’s right – there will be 4 axioms left, because the axiom is accepted without proof, and if it has been proven within the framework of this theory, then it is not an axiom, but another conclusion.
Of course, this will not affect objective reality in any way and will not change the basis of the universe. Our set of axioms will simply change, and this will happen only within the framework of geometry. Because an axiom is an axiom only within the framework of its own theory, and outside its boundaries it can be an axiom, a conclusion, and even, as mentioned above, a deliberately false idea.
And this property of axioms, which requires that they not be provable within the framework of their own theory, is very useful in practice.
Decompose into axioms
So, let’s imagine that we have axioms:
brick
building mixture
floor slabs
beams for door and window openings
You can build this house from scratch on site, or you can build two halves and then move them towards each other.
But no matter how this house was built, it can be decomposed into axioms in only one way.
Because if you split the house into two different sets, where one has fewer bricks but more beams, then they are interchangeable and one can be assembled from the other.
Which contradicts the essence of axioms as units that cannot be proven.
It’s like decomposing numbers into prime factors; if you managed to decompose in more than one way, then the results of the decomposition indicate not only prime numbers.
Although, this is not the most successful example, because in the case of factorization, this uniqueness is two-sided. But when we have a more complex theory, then there is no uniqueness in the opposite direction.
We can take the text, discard all whitespace characters, convert it to lower case and decompose it into printable characters, writing the result in the format “a#2;b#1;c#54;d#92;z#23;”. And for each text this decomposition will be the only possible one.
For the line “mom washed the frame” it would be: “a#4;l#1;m#4;p#1;u#1;s#1;”
For the article “Reality exists and this must be taken into account”: “!#1;”#80;%#32;(#10;)#10;*#1;,#275;-#64;.#182;/#63;:#34;=#2;? #9;@#2;[#11;]#11;_#7;a#43;b#21;c#37;d#35;e#39;f#11;g#12;h#33;i#67;k#15;l#19 ;m#27;n#37;o#36;p#41;q#1;r#41;s#42;t#72;u#25;v#8;w#11;x#1;y #7;z#1;»#1;a#1105;b#209;c#565;g#179;d#365;e#1119;f#117;z#267;i#1350;y#172 ;k#497;l#466;m#538;n#1043;o#1602;p#376;r#733;s#802;t#1070;u#269;f#83;x#166;ts #76;h#336;sh#62;sh#63;ъ#5;y#352;b#281;e#73;yu#83;0#52;i#271;е#70;№#4 ;1#41;2#57;3#19;4#15;5#9;6#7;7#16;8#15;9#9;”
For the article “The UNITS Paradigm to the Masses”: “!#3;”#36;##3;%#5;(#11;)#11;*#7;,#155;-#15;.#118;/#34;:#8;? #8;@#3;#4;^#2;_#4;a#14;b#7;c#15;d#13;e#13;g#2;h#11;i#10; j#2;l#3;m#16;n#19;o#15;p#4;r#11;s#16;t#21;u#9;v#1;x#1;y# 2;“#8;”#8;a#732;b#125;c#403;d#151;d#249;f#766;f#86;z#191;i#806;y#111; k#273;l#337;m#418;n#645;o#1102;p#257;r#455;s#562;t#796;u#241;f#42;x#85;ts# 25;h#215;sh#35;sch#58;ъ#4;ы#194;ь#231;e#47;yu#47;i#153;ё#42;0#31;‑#4; —#2;“#3;„#3;1#10;2#21;3#8;4#7;5#9;6#12;7#10;8#2;9#5;”
And in addition to checking the parentheses for pairing, this gives us another output. If decomposition into axioms gives a single result, then the discrepancy in the decomposition indicates that different texts were decomposed.
Yes, in fact, as a result of decomposition into axioms, a checksum is obtained:
if it differs, then we compare different texts
if it matches, then it can be either the same or different texts
we don’t know what the original text was
Yes, we can try to overcome the meaninglessness of the result by breaking down the text not into axioms in the form of letters, but by focusing on an intermediate option – parsing the text into meaningful phrases. However, we will have to pay for this by the variability of decomposition into word combinations, which means we will not be able to use this as a control mechanism. Because the decomposition into phrases can not only be different for different people, but also different for the same person at different times.
And since there is variability in the decomposition, there is no way to use a checksum to check the immutability of the text.
And it turns out that we have two options:
Completely lose the meaning of the text, but reliably (but not 100%) determine its immutability.
Or keep an indication of the meaning of the text, but completely lose the mechanism for controlling its immutability.
And those who read the previous articles in the series (https://habr.com/ru/articles/776550/ and https://habr.com/ru/articles/777992/) and the comments to them have already guessed what they were talking about.
7 axioms of the International System of Units (SI)
Yes, we are talking about point 2 in the UNITS Paradigm – “Values must have a dimension corresponding to their physical meaning” and the example of the dimension of liquid viscosity.
We have three viscosity size options:
the original “three-story” dimension, fully consistent with the formula and physical meaning of liquid viscosity.
option “Pass” which makes sense within the framework of a specific mathematical apparatus using tensor calculus.
option in SI – “m-1kg1s-1”
Clause 1 allows you to remind the operator of what he is dealing with and how to use the formula correctly.
Clause 2 is useful in certain situations, but the original meaning has already been lost, and there is no unambiguous “checksum” yet.
Clause 3 allows you to insure yourself against incorrect addition. The meaning is completely lost, but the only possible option for reducing the dimension in the form of expansion to 7 axiomatic base SI quantities gives us a “checksum”.
Yes, point 3 does not provide 100% protection against the fact that quantities with different physical meanings will be added. Because values with different meanings can give the same abbreviation.
For example, “surface tension” and “energy exposure” are shortened to variant “kg1s−2”. And purely mechanical “N/s” and spectral flux density “(W/m^2)/m” reduced to the same option “m1kg1s−3”.
And yet, despite all the collisions and sometimes surprising variations of the resulting checksums, reducing the dimension to 7 basic SI values is a reliable tool for controlling the coincidence of dimensions in the process of computer calculation.
And precisely because the basic SI quantities are axioms – designated units that cannot be converted into each other, and therefore have a single reduction option.
And all this becomes possible after we abandon the strange idea that axioms are the real truth, and return to what we were taught at school: “an axiom is a position accepted without proof, that is, by an effort of will.”
Seventh SI unit
But there is also a fly in the ointment for this barrel of honey, which I poured on the axioms so that they become more attractive, understandable and useful for you (and this is really useful, to understand what the axiomatics of your own model, theory or construction are, and how to deal with this very axiomatics work).
The fact is that physicists, unlike mathematicians, were able to deduce one axiom from others. The temperature unit “Kelvin” is already given by recalculating through a constant into “Joule”.
This means that, scientifically, temperature should be measured not in “Kelvins” or “degrees Celsius”, but in “kg1m2s−2”. Live with it now.
And don’t try to find it in the option “kg1m2s−2” physical meaning, because this is simply the only option for decomposition into axioms – it is convenient, but meaningless.
And this also means that, when reducing the dimension in the process of computer calculation, “Kelvin” should also be recalculated into “J,” but exclusively as a dimension, without affecting the numerical value of the calculation results.
In general, there are a lot of questions and let’s postpone this topic until after the New Year.
