Why science must use math -- feedback welcome -- work in progress

Incorrect. As already explained, theta is the angle between the directions of the vectors. The interior angle of a triangle formed by placing the vectors head-to-toe is \(180^{\circ} \; - \; \theta\) and \(\cos \theta \quad = \quad - \cos \left( 180^{\circ} \; - \; \theta \right) \).

It's obvious that you are incorrect because if a = b and the angle between their directions is zero and you predict the magnitude of final vector = \(\sqrt{a^2 + a^2 - 2 a^2} = 0\).

You are off base in other respects, since in every post answering you in this thread, I have made it clear that you used the word stationary which has a well-defined meaning in Newtonian models and no well-defined meaning in Lorentzian models. That you now, as I anticipated, want to claim that something bogus results from using the Newtonian model does not make internal mathematical sense, because the Newtonian model is incompatible with the axiom that no two objects have a relative velocity larger than c.

If you accept that as a postulate, then the addition of that axiom to the Newtonian model is internally inconsistent (because math is fragile) and must be rejected. Because c is large compared to human relative velocities, the unembellished Newtonian model at best could be an approximate physical theory for collections of low-speed objects -- that would be limiting it's domain of applicability.

 

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Yes, you are right, because you have specified "the angle between their directions" (of the vectors). I was careless. I consider the angle between a and b taking into account the graphics.

And is as I said.
You cannot make a gathering of two vectors, taking into account the SR.

It is a difference between us. I consider the mathematics is the most rigorous, accurate and precise science.

 

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I already did.

Like I said, the expression is:
\(c \; \tanh \, \cosh^{\tiny -1} \, \left( \cosh \, \tanh^{\tiny -1} \, \frac{a}{c} \; \cosh \, \tanh^{\tiny -1} \, \frac{b}{c} \; + \; \cos \theta \; \sinh \, \tanh^{\tiny -1} \, \frac{a}{c} \; \sinh \, \tanh^{\tiny -1} \, \frac{b}{c} \right)\)
and you can get this by doing just one term of the matrix multiplication described.

Using (hyperbolic) transforms and assuming a and b are non-negative sublight speeds, we have the following identities:
\( f(x) = \tanh \, \cosh^{\tiny -1} x = \sqrt{1 - x^{\tiny -2}} \\ g(x) = \cosh \, \tanh^{\tiny -1} \, x = \frac{1}{ \sqrt{1 - x^{\tiny 2}}} \\ h(x) = \sinh \, \tanh^{\tiny -1} \, x = \frac{x}{ \sqrt{1 - x^{\tiny 2}}} = x g(x) \)

Thus
\(c \; \tanh \, \cosh^{\tiny -1} \, \left( \cosh \, \tanh^{\tiny -1} \, \frac{a}{c} \; \cosh \, \tanh^{\tiny -1} \, \frac{b}{c} \; + \; \cos \theta \; \sinh \, \tanh^{\tiny -1} \, \frac{a}{c} \; \sinh \, \tanh^{\tiny -1} \, \frac{b}{c} \right) \\ = c \, f \left( g \left(\frac{a}{c} \right) g \left(\frac{b}{c} \right) + h \left(\frac{a}{c} \right) h \left(\frac{b}{c} \right) \, \cos \theta \right) \\ = c \, f \left( g \left(\frac{a}{c} \right) g \left(\frac{b}{c} \right) \left( 1 + \frac{ab \, \cos \theta}{c^2} \right) \right) \\ = c \, f \left( \frac{c^2 + ab \, \cos \theta}{c^2 \sqrt{ \left( 1 - \frac{a^2}{c^2} \right) \left( 1 - \frac{b^2}{c^2} \right) }} \right) \\ = c \, \sqrt{ 1 - \frac{ \left( c^2 - a^2 \right) \left( c^2 - b^2 \right) }{\left(c^2 + ab \, \cos \theta \right)^2 }} \\ = \frac{c}{c^2 + ab \, \cos \theta } \, \sqrt{ c^4 + 2 a b c^2 \cos \theta + a^2 b^2 \cos^2 \theta + a^2 c^2 + b^2 c^2 - a^2 b^2 - c^4 } = \frac{\sqrt{ a^2 + 2 a b \cos \theta + b^2 - \frac{ a^2 b^2}{c^2} \sin^2 \theta }}{1 + \frac{ab}{c^2} \, \cos \theta } \)

This last expression, simplifies under various assumptions, always assuming a and b are non-negative and sub-light.
\( \begin{eqnarray} a = 0 & \Rightarrow & \frac{\sqrt{ b^2 }}{1} = b \\ b = 0 & \Rightarrow & \frac{\sqrt{ a^2 }}{1} = a \\ c \to \infty & \Rightarrow & \sqrt{ a^2 + 2 a b \cos \theta + b^2 } \\ \theta = 0 & \Rightarrow & \frac{\sqrt{ a^2 + 2 a b + b^2 }}{1 + \frac{ab}{c^2} } = \frac{ a + b }{1 + \frac{ab}{c^2} } \\ \theta = 180^{\circ} & \Rightarrow & \frac{ \left| a - b \right| }{1 - \frac{ab}{c^2} } \\ \theta = 90^{\circ} \quad \textrm{or} \quad \theta = 270^{\circ} & \Rightarrow & \sqrt{ a^2 + b^2 - \frac{ a^2 b^2}{c^2} } \end{eqnarray}\)

 

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Calculations are ingenious and accurate (at least I found no errors).
This is the scalar. Can you say something about the orientation?
Is somewhere a link about them?

 

I don't think you have the math to deal with orientation, since in hyperbolic geometry, large triangles need not have their internal angles sum to \(180^{\circ}\).

In the limit of very small velocities, you can make an equilateral triangle with \(\theta \, = \, \frac{2 \pi}{3} \quad ( \textrm{internal angle:} \; 60^{\circ} )\) just as you can in Euclidean geometry.
With \(a \, = \, b \, = \, \sqrt{\sqrt{8} - 2} \,c \; \approx \; 0.91018 \, c\), you can make an equilateral triangle with \(\theta \, = \, \frac{3 \pi}{4} \quad ( \textrm{internal angle:} \; 45^{\circ} ) \)
With \(a \, = \, b \, = \, \sqrt{\sqrt{\frac{16}{3}} - \frac{4}{3}} \,c \; \approx \; 0.98796 \, c\), you can make an equilateral triangle with \(\theta \, = \, \frac{5 \pi}{6} \quad ( \textrm{internal angle:} \; 30^{\circ} ) \)
With \(a \, = \, b \, = \, \sqrt{ \sqrt{48} + \sqrt{24} - \sqrt{8} - 8} \, c \; \approx \; 0.99938 \, c\), you can make an equilateral triangle with \(\theta \, = \, \frac{11 \pi}{12} \quad ( \textrm{internal angle:} \; 15^{\circ} ) \)

 

That’s not what I meant. I meant that since it is so expensive and timing consuming aesthetic mathematical structures are taking on greater importance in theoretical physics.

“Similarly, Eugene Wigner said that the unreasonable efficiency of mathematics is "a wonderful gift which we neither understand nor deserve." Thus we have a problem that may seem too metaphysical to be addressed in a meaningful way: Why do we live in a comprehensible universe with certain rules, which can be efficiently used for predicting our future?”- ANDREI LINDE

You’re welcome. I doubt that we have the same taste but I also enjoyed this year's Edge Question.

“Scientists' greatest pleasure comes from theories that derive the solution to some deep puzzle from a small set of simple principles in a surprising way. These explanations are called "beautiful" or "elegant". Historical examples are Kepler's explanation of complex planetary motions as simple ellipses, Bohr's explanation of the periodic table of the elements in terms of electron shells, and Watson and Crick's double helix. Einstein famously said that he did not need experimental confirmation of his general theory of relativity because it "was so beautiful it had to be true."

WHAT IS YOUR FAVORITE DEEP, ELEGANT, OR BEAUTIFUL EXPLANATION?

 

Nevertheless, Field and Balaguer are at the centre of the present discussion about the status of maths for science. So, if your title is accurate, you'll need to deal with their position, if you're intending to write a serious article.

Okay, but in that case, your claim appears to be false by counter-example. What maths was required for Darwin to introduce his theory of evolution?

High profile free will deniers include Einstein, so your claim here is also false by counter-example.

Neither evolution nor global warming deniers are proposing advancements to science, as far as I'm aware, so I dont see how they're relevant.
Generally, it's not clear whether you intend to support the claim implied by your title or whether you intend some argument on the lines of this:
1) there is no new science without maths
2) various people make claims of new science on the internet
3) if there is no maths in these claims, then there is no new science.
As it stands, you haven't gone anywhere towards supporting the first claim and the second seems too trivial to justify the kind of article you appear to be planning. In any case, it would be helpful if you were to state clearly what your article is intended to support, who your intended audience is, etc.

 

Darwin speaks of correlations of heretibility, relative fertility and differential reproductive success in chapters 4,5 and 8 of Origin of Species just to list a few examples. Not only are these mathematical observations and inferences, but they have precision that lead these statements to be tested.

 

All three of those chapters are in plain text. He symbolises some terms for convenience, but if that constitutes maths, then it is very difficult to imagine that there is anyone, capable of writing in their native language, who is incapable of maths. What exactly do you mean by "math", in your title?