# Einstein versus Pythagoras

The following diagram conveys the general notion of relativity. This is not Einstein’s relativity due to the constant speed of light, but it is useful to convey the notion of relativity in general.

The diagram gives an imaginary case of the Moon circling the Earth such that Earth and Moon do not rotate themselves but are always in the same orientation to the same distant stars. The Blue and Red dots are observers who remain oriented to those distant stars. An observer on Earth at the Blue dot would be able to see all sides of the Moon – assuming that Earth were transparent. For observer Blue the Moon is rotating, even though it isn’t with respect to the distant stars. If the Moon is on the left hand side, Blue will see its right side. If the Moon is on the right hand side then Blue will see its left side. Similarly for top and bottom. Thus, what actually is fixed is observed as rotating. Or, if the Moon were actually rotating and Blue not, subtract one seeming rotation to eliminate this observation effect.

Moon circling the Earth while none are rotating themselves

The principle of relativity may also be explained by comparing a car driving past a house. For the observer in the house, the car has a speed. However, seen from the position of the driver, it is the house that passes by at that speed. This example also conveys that observation is relative to the position of the observer. In this case, however, the example is not that strong. The car has a brake, and the house hasn’t. Thus in this case it makes more sense to say that it is the car that is causing the speed difference.

A person who turns his head sees the universe spinning around him or her. It doesn’t make much sense to hold that everything is relative and that the universe is spinning around with close to infinite speed and energy. Though it would be difficult to locate, the center of the universe is a more logical point to describe events from.

Above diagram doesn’t have the complications of a car brake or the turning of your head or Einstein’s use of the constant speed of light. It shows observational relativity in terms of logic. Though the Moon does not rotate itself (Red is always oriented at the same distant stars) it seems to rotate for Blue (with the same orientation).

#### Pythagoras and the definition of space

Let me quote from my book Conquest of the Plane p85:

There is the paradoxical situation that we may take great pains to prove something that from another point of view is merely a matter of definition. The Pythagorean Theorem is commonly expressed in terms of sides a, b and c. For the circle c = r. Then we get:

♦ Pythagoras convinces us that we have to prove that c^2 = a^2 + b^2
♦ For a distance we now define that c^2 = a^2 + b^2

The solution to this paradox is that Euclid used other axioms than we now do for the distance. Though Pythagoras (ca. 572 – 500 BC) lived before Euclid (around 300 BC), we can say in a figure of speech: Given the Euclidean axioms Pythagoras has to prove his Theorem. Once he got the proof he could define the circle. Without the proof he might define the circle but then would have to prove that it really exists. That said, in analytic geometry it is easier to work the other way around. Starting with formulas is a fast way to get up and running. Using distance we can define parallel lines as lines that have equal distance. With distance the circle arises naturally. The notion of distance is crucial for the Euclidean plane. We surmise that Euclid relied on a notion of distance too by using the compass.

What remains in all this is our notion of Euclidean space: a notion of straightness of lines and flatness of the plane that might derive from everyday experience but that essentially is a concept of the mind, and essentially a definition.

What you should take away from this is: the definition of “space” is Euclidean space. If you think about “space” then this is what you think. You cannot change what already has been defined to generate your understanding.

#### Einstein’s historical context

As observational relativity because of the constant speed of light causes measurement errors, Einstein eliminated those errors by adapting “space”. But can you change the notion of space if it already has been defined by Pythagoras and Euclid ? An elegant way to deal with systematic measurement errors doesn’t change “space”. Something else is happening here.

Let me quote from Conquest of the Plane p195 that describes Albert Einstein’s historical context.

A key issue in the theory of science is the issue of measurement. Physics before Newton suffered huge losses in intelligence, time and energy to discussions on unobservables and metaphysics. This in fact lasted partly into the 19th century with discussions on the ‘ether’. Their solution was to put a stop to fruitless discussion and concentrate on what can be measured. You don’t know what it is, but it moves this way, at that speed, and if you hit it here, then it moves there. This technical approach worked wonders, though it still seems that some theorists assume some ‘whats’ to derive their theories on the ‘hows’ (as Bohr’s atom model).

(…) A key notion below will be that physics might ‘overshoot’ by concentrating on measurement and by neglecting definitions and logic.

(…) Einsteins model subsequently seems to confuse the definition of space, given by the definitions of Euclid, and empirical space as measured by the instruments of physicists.

(…) Modern physicists shy away from the possibility that space and time have independent definitions within the mathematical modelling of the world. They regard space and time as what they measure. However, they don’t seem to see that they can be hopelessly confused when they measure speed in meters / second while those meters and seconds change under measurement. My impression is that it is better to accept measurement error and try to explain that error.

#### Education in mathematics vs physics

Please observe that I am no physicist and rely for that on what I remember from gymnasium. The above is a view from the position of the education in mathematics. The views from the education in physics may be different. There may be relativity in education as well.

The above concerns a minor comment in COTP. Its real contribution lies elsewhere. PM. COTP also allows the earlier discussion of derivatives, so that physics education can start using those much earlier too.

The issue might be resolved empirically. A physicist would have to show that it is impossible to describe the measurement error in Euclidean space, so that the use of Riemann curvature is not just a historically understandable way of modelling but also necessary. It would be more interesting of course to see that the Riemann form generated other confusions.

Edward Frenkel holds that the Pythagorean Theorem meant the same to people 2500 years ago as it means to people nowadays. This doesn’t seem true to fact, though of course is hard to prove. At least the above shows that we have added shades of understanding that were lacking in the past. Some historians hold that Euclid did not present a cosmology or theory of space but a theory of measuring. However, it seems that the latter presupposed the first, see point (v) here. Also, Frenkel emphasizes the importance of the Riemann model, and thus should admit that modern physicists claim another view of space than Pythagoras and Euclid, so that  he cannot uphold that “sameness”. Overall, Frenkel is a research mathematician and has no background in the empirical science of education, so he is producing a lot of nonsense. More on that later on.