But Alex:
Definitions can be wrong. They can be contradictory. If you 'define' "the moon that circles the earth' as "the moon that circles Pluto" you have made a false definition.
Definitions may also be partial. The claim "1 + 1 is defined as 2" is contradictory and impossible; or it is true by agreement, but only a partial view of the situation. In fact, the very statement "1+1=2" admits that there ARE two '1's;
but fails to note that they are two DIFFERENT '1's.
Thus "this 1" and "that 1"; and though there are TWO '1's' it is only a partial description of the situation because it doesn't acknowledge their difference: there is still only one "this" and one "that". There are not two 'this's'; there are not two 'that's'.
Sure I studied University-level math; but math involves many assumptions that people tend to take for granted without question.
No amount of definition will make contradictions true. You can not multiply a number by itself; only by a different number. To multiply 'a' by 'a' involves 'this a' and 'that a' (if it didn't, there would only be one 'a' to talk about).
That is why the square root of minus 1 is no big deal: the two numbers that square to give a third, are always different. "This 2" times "that 2" gives 4; but not 4 "this's" or 4 "that's"; only "4" from a "2 GENERALISATION" perspective. Math is really about quantum jumps in perspective; or exchanges of perspective; so is physics. Relativity and quanta.
Only a generalisation perspective allows one to say that a vector can be multiplied by itself: strictly speaking: to do any operation like 'multiply' or 'divide' or 'add' or 'subtract' requires TWO items. Those items may be the same in some ways; but cannot be the same in EVERY way; or you would have only one item.
You can not make contradictions true by definition. You can not define Pluto's moon as Earth's moon. There are two moons here; but only from the generalisation perspective "moon". Strictly; there are not two WHOLE items here; only TWO PARTIAL derivatives. From a wholeness-perspective: there is actually just one 'earth moon' and one 'pluto moon' here.
The following is rather speculative:
Lorentz transformations involve interchanges of perspective.
A hypercube (4D cube) can be viewed from 8 perspectives in 3D-viewing mode. Effectively you have the option of rotating the implied seven 3D-cube-'walls' of the hypercube one after the other into an eighth 'wall' 3D-view you get from 3D-land.
Any one view of the hypercube gives you a 3D-cube in 3D-land, plus 7 'time cubes' (reference cubes). Say 3 'pasts' and 4 'futures'.
If you look at the hypercube in 4D hyper-space; and it is travelling at c; it will look like as it looks from 3D cube-space. This is because of Lorentz contraction of dimension at c.
Similarly; in 3D-land; a 3D-cube moving at c is contracted into the appearance it would have in 2D 'flat-land'; looks like a square.
A Hypercube moving at c-squared in 4D-hyperspace will look to a hyperspace observer like a square (as it looks in flatland, two dimensions down).
Thus e = mc-squared takes on a new perspective.
Looks like 'e' is about 'future'; 'm' is about 'past'.
Regarding the hypercube moving at c in hyperspace: the collapsed dimensions; the pasts and futures of the 3d-'walls' will cancel (the fourth future cancels with the initial view (a past): it will be 'frozen in hyper-time' at hyperspace-c.
Futures repel (only one future can occur in any choice of two futures; pasts attract (as new past-combinations creates new future option (hence future-options of space expand at accelerating rate).
Like charges repel: you can't get anywhere with a 4past/3future and a 4past/3future combination (unless it breaks down to a lower dimension collection- as represented in sub-loops of Feynman diagrams). ("Line-world" 1D objects are a dimensional sub-loop to "flat-land" 2D objects are a dimensional sub-loop to 3D-cube-land objects are a dimensional sub-loop to 4D space-time (hyperspace) objects). (Schrodinger wave function/ Feynman sum of histories might thus be explained).
Unlike charges attract: you can get somewhere with a 4past/3future and a 3past/4future combination: you can cancel those two and end out with 7 past/7 future thus a 'present'. Find another 'present' and a new future; a new combination becomes possible (and you just expanded hyperspace).
Then one might suppose one has a new 'rung in the ladder of inter-dimensional 'DNA''; and the expansion of space provides another 3/4 combination looking for a match to build some more Universe-world-line with.
The forces of nature will probably prove Dirac's intuition to be true: he noticed that the ratio of the weakness of gravity relative to the other forces was much the same as the ratio between the distance across an atom and across the Universe. Forces will probably be found to be ratios. "c" is known to be a ratio; maybe "c" can be shown to be "pi".
There seems to be a kind of 'Lorentz musical chairs' exchanges game going on in nature.
You may have guessed I've been reading that amazing book: "Hyperspace" by Michio Kaku. Got new ideas thinking about diagram on page 73, for example (projection of cube onto flat-land; of hypercube on to 3D-land).
-dolphin |