Comment by adrian

There are many other kinds of ratios.

Ratios of collinear vectors are scalars a.k.a. "real" numbers, ratios of other kinds of vectors are matrices, ratios of 2D-vectors are "complex" numbers, ratios of 2 voltages are scalars a.k.a. "real" numbers, and so on.

In general, for both multiplication and division operations, the 3 sets corresponding to the 2 operands and to the result are not the same.

Only for a few kinds of multiplications and of divisions the 3 sets are the same. This strongly differs from addition operations, which are normally defined on a single set to which both the operands and the result belong.

In practice, multiplications and divisions where at least one operand or the result belong to another set than the remaining operands or result are extremely frequent. Any problem of physics contains such multiplications and divisions.

An algebraic structure of division algebra is not at all necessary for a division operation to exist.

On the contrary, there exists a huge number of kinds of division operations, but only very few division algebras.

For any kind of multiplication operation you can define an associated division operation (or 2 associated division operations, for non-commutative multiplications), which accepts as input the product as the dividend and one factor as the divisor, and which returns the other factor as the quotient.

In general, the division operation can never be defined for all pairs of product with factor for which the multiplication is defined, not even in a division algebra, where division by zero remains undefined. Nevertheless, in almost all cases a division operation can be defined for many of the possible product-factor pairs.

While there are some vector pairs for which the division operation is undefined, for most vector pairs the division operation is perfectly defined. Given such a pair of vectors there exists a unique matrix that will produce the dividend vector when multiplied with the divisor vector. Every kind of division is the inverse of some kind of multiplication, and this kind of division is the inverse of the matrix-vector multiplication.

The division of vectors is quite important in practice, for example for finding the rotation that will transform one object into another object, which is identical, but rotated.

Any physical measurement is a division that is not performed in a division algebra, because the operands belong to the set of values of the measured physical quantity, while the result belongs to a different set, i.e. it is a rational number that approximates the scalar a.k.a. "real" number that corresponds to the measured value.

Saying that a measurement is a division operation is not some kind of metaphor, but it is a concrete description of how to implement it. For example, there are a very large number of kinds of analog-digital converters, which are used for voltage measurement. Because they must compute the division between the measured voltage and a reference voltage, each of the many possible structures (e.g. flash converters, successive-approximation converters, pipelined converters, cyclic converters, ramp converters, sigma-delta converters, etc.) corresponds to one of the algorithms that can be used for computing a division, even with pen and paper.

All the useful parts of mathematics are not isolated of the physical world, but they abstract the properties of the physical world, to enable the description of the physical world by models about which one can reason formally.

What are called now "real numbers" were called "measures" by the ancients (who called "numbers" only the values that can be obtained by counting). The theory of "measures", i.e. "real numbers", has not been conceived in a dream, separated from the physical world, but it was conceived as an axiomatic model that attempted to match the perceived behavior of the physical quantities, such as lengths.

Similarly all the other valuable mathematical concepts have been conceived for using them in mathematical models of the physical world.

Once a useful axiomatic framework existed, there have also been many mathematical researches that were motivated only by intellectual pleasure disconnected from immediate practical necessities, like in the case of many results in number theory, or in the study of systems where only less axioms are used or some axioms are replaced, in comparison with any system applicable in practical mathematical models.

However, completely "pure" mathematical research, where one would study the consequences of some random axioms completely unrelated to the physical world (i.e. which would not use any of the mathematical concepts abstracted from the physical world, e.g. numbers, topologies etc.) has been very rare and it has not produced interesting results.

jiggawatts 3 days ago

> Ratios of collinear vectors are scalars a.k.a. "real" numbers, ratios of other kinds of vectors are matrices,

Wat? Vectors do not form a division algebra. You can't "divide" vectors! Perhaps you're thinking of the dot product, which returns a scalar.

Also, you're conflating physical/engineering concepts (such as units of measure) with mathematical abstractions such as number spaces, which don't have units.

Physical measurements exist in the real world, with its limitations, units, and practicalities.

Mathematical numbers exist in a pure theory space that is completely and totally independent of any physical reality. They're just axioms and rules. Definitions and logical conclusions.

Don't mix up the two!

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adrian_b 2 days ago

An algebraic structure of division algebra is not at all necessary for a division operation to exist.
On the contrary, there exists a huge number of kinds of division operations, but only very few division algebras.
For any kind of multiplication operation you can define an associated division operation (or 2 associated division operations, for non-commutative multiplications), which accepts as input the product as the dividend and one factor as the divisor, and which returns the other factor as the quotient.
In general, the division operation can never be defined for all pairs of product with factor for which the multiplication is defined, not even in a division algebra, where division by zero remains undefined. Nevertheless, in almost all cases a division operation can be defined for many of the possible product-factor pairs.
While there are some vector pairs for which the division operation is undefined, for most vector pairs the division operation is perfectly defined. Given such a pair of vectors there exists a unique matrix that will produce the dividend vector when multiplied with the divisor vector. Every kind of division is the inverse of some kind of multiplication, and this kind of division is the inverse of the matrix-vector multiplication.
The division of vectors is quite important in practice, for example for finding the rotation that will transform one object into another object, which is identical, but rotated.
Any physical measurement is a division that is not performed in a division algebra, because the operands belong to the set of values of the measured physical quantity, while the result belongs to a different set, i.e. it is a rational number that approximates the scalar a.k.a. "real" number that corresponds to the measured value.
Saying that a measurement is a division operation is not some kind of metaphor, but it is a concrete description of how to implement it. For example, there are a very large number of kinds of analog-digital converters, which are used for voltage measurement. Because they must compute the division between the measured voltage and a reference voltage, each of the many possible structures (e.g. flash converters, successive-approximation converters, pipelined converters, cyclic converters, ramp converters, sigma-delta converters, etc.) corresponds to one of the algorithms that can be used for computing a division, even with pen and paper.
All the useful parts of mathematics are not isolated of the physical world, but they abstract the properties of the physical world, to enable the description of the physical world by models about which one can reason formally.
What are called now "real numbers" were called "measures" by the ancients (who called "numbers" only the values that can be obtained by counting). The theory of "measures", i.e. "real numbers", has not been conceived in a dream, separated from the physical world, but it was conceived as an axiomatic model that attempted to match the perceived behavior of the physical quantities, such as lengths.
Similarly all the other valuable mathematical concepts have been conceived for using them in mathematical models of the physical world.
Once a useful axiomatic framework existed, there have also been many mathematical researches that were motivated only by intellectual pleasure disconnected from immediate practical necessities, like in the case of many results in number theory, or in the study of systems where only less axioms are used or some axioms are replaced, in comparison with any system applicable in practical mathematical models.
However, completely "pure" mathematical research, where one would study the consequences of some random axioms completely unrelated to the physical world (i.e. which would not use any of the mathematical concepts abstracted from the physical world, e.g. numbers, topologies etc.) has been very rare and it has not produced interesting results.

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Comment by adrian_b