B Why is there so much multplication in physics?

[dainceptionman_02]

i notice with physics, all the equations are a bunch of scrambled sentences or variables being multiplied together. why the multiplication for everything?

 

[fresh_42]

i notice with physics, all the equations are a bunch of scrambled sentences or variables being multiplied together. why the multiplication for everything?

Can you give an example?

It is generally the case that we can only add quantities of the same kind: length + length or velocity + velocity. However, we can always multiply quantities regardless of their nature: length / times, mass * acceleration, force / area. These multiplied quantities make sense in a certain context, but adding them won't.

 

[topsquark]

i notice with physics, all the equations are a bunch of scrambled sentences or variables being multiplied together. why the multiplication for everything?

$s = s_0 + v_0 t + \dfrac{1}{2} a t^2$

$v = v_0 + at$

$I = I_{CM} + Md^2$

Yes, there's a lot of multiplication, but there's also a lot of addition.

I'm going to go out on a bit of a limb here, but one possible way to see this is that Physics tries to deconstruct things into more basic principles. The example I'm going to use is ridiculously oversimplified (and, frankly, backward) but it will illustrate the point.

Look at velocity. It has units of m/s. We can "reduce" this to two more basic concepts: distance and time. Skipping some Mathematical details, we divide distance by time, which is essentially multiplication. The result is that we have described velocity by two more basic concepts. We are separating the concepts by separating the units, and the only way to do that is by multiplication.

If you want to look at a better (but more complicated) example of this, take a look at how we break down the concept of moment of inertia from the torque-angular acceleration equation.

Mind you, this is hardly the whole story, but it should give you a better idea of why the multiplication happens.

-Dan

 

[kuruman]

Yes, there's a lot of multiplication, but there's also a lot of addition.

Actually, there is a quite a bit of addition, much more than multiplication, but one tends not to think about it. For example, when one writes $p=mv$ keep in mind that $mv=\sum_{i=1}^{N}m_i v_i$ where $N$ is a humongous number for, say, a 1-kg ball that has $N$ atoms. We tend to forget that multiplication is a quick way to add things.

 

[Vanadium 50]

You obviously think there is too much multiplication. What would be just the right amount?

 

[jbriggs444]

why the multiplication for everything

Because lots of things are proportional to lots of other things? And because units?

 

[Chestermiller]

i notice with physics, all the equations are a bunch of scrambled sentences or variables being multiplied together. why the multiplication for everything?

This is what. you spend your valuable time thinking about?

 

[A.T.]

You obviously think there is too much multiplication. What would be just the right amount?

We need equal representation of all operations. This multiplication supremacy has lasted long enough.

 

[fresh_42]

We need equal representation of all operations. This multiplication supremacy has lasted long enough.

More power to addition!
$$
\exp(x+y)
$$
...
...
...
$$
\exp(x) \cdot \exp(y)
$$

Oops.

 

[Ibix]

More power to addition!
$$
\exp(x+y)
$$
...
...
...
$$
\exp(x) \cdot \exp(y)
$$

Oops.

Seems $\ln$ical.

 

[dainceptionman_02]

for example, in electricity and magnetism, you have Coulomb's Law, various electric fields, potential difference, capacitance, resistivity, resistance, various magnetic forces and magnetic fields. all of them have tons of things being multiplied together. everything is multiplication! i know that unit wise, to get tesla, you have to do dimensional analysis with a bunch of variables, but i just always wondered why the theme of physics equations is always a bunch of products...

 

[Ibix]

i just always wondered why the theme of physics equations is always a bunch of products...

Take $F=ma$. You know that the force needed to accelerate something increases for bigger masses. For a fixed mass you need a bigger force to get a bigger acceleration. Just from those two constraints, and no other reasoning, mathematically you must have $F=ma$ or $F=m+a$. But the latter makes no sense - if you have zero force then you have $m=-a$ and an expectation that anything with mass accelerates when left alone (and if you put the vector signs in it makes even less sense). So $F=ma$. You can apply similar reasoning to other things - in your Coulomb's law example you need your charges and your inverse square to multiply or you get silly answers.

And the dimensional analysis is an interesting one. Take $F=m+a$ again. What does it even mean to add a mass to an acceleration? I know what 1g + 2g is, and I can even work around weird units like 1kg + 3lb. But what is one gravity more than a kilogram? Or one meter more than one Celsius? Products, though, make sense, like length times length being area. I think this is primarily because units themselves are multiplications. Formally something that has a mass of seven grams is something whose mass is seven times some agreed standard mass we call a gram.

 

[fresh_42]

for example, in electricity and magnetism, you have Coulomb's Law, various electric fields, potential difference, capacitance, resistivity, resistance, various magnetic forces and magnetic fields. all of them have tons of things being multiplied together. everything is multiplication! i know that unit wise, to get tesla, you have to do dimensional analysis with a bunch of variables, but i just always wondered why the theme of physics equations is always a bunch of products...

Dimension is the key! In its physical as well as in its mathematical sense. A dimension describes a physical quantity that is mathematically a vector space: we can add and stretch. Every other operation leaves the dimension. However, physics is the attempt to understand nature, and nature is full of different dimensions. If we consider how one changes if we change another, then we get a comparison: a proportion or exponentiation. Exponentiation is always related to growth, but proportions are all over the place, and the only possibility to compare two different dimensions! Voltage and current in the same lamp are related by power and resistance, a proportion. Weight and mass are related by the planet, a proportion. Proportions are actually the seed of science: the major subject in classical geometry. We cannot add current and voltage, but we can compare two relations of current times voltage.

 

[Vanadium 50]

for example, in electricity and magnetism, you have Coulomb's Law...

Even these examples don;t work. Two forces on an object, you add them. Resistors in series? You add them Capacitors in parallel, you add the,m.

 

[DrClaude]

We need equal representation of all operations. This multiplication supremacy has lasted long enough.

Multiplication is nothing but repeated addition, so it is addition all the way down!

 

[fresh_42]

Multiplication is nothing but repeated addition, so it is addition all the way down!

As a fan of algebras, I seriously have to disagree.

 

[DrClaude]

As a fan of algebras, I seriously have to disagree.

Ah! You mathematicians with your fancy algebras... Time to remove my tongue form my cheek.

 

[PeroK]

Multiplication is nothing but repeated addition, so it is addition all the way down!

Except when multiplication is non-commutative!

 

[fresh_42]

Ah! You mathematicians with your fancy algebras... Time to remove my tongue form my cheek.

If you're not good we take back our differential operators again, or to make some promotion, our alternators.

 

[martinbn]

Except when multiplication is non-commutative!

Even with commutative multiplication, it is not repeated addition. How is $f(x)g(x)$ a repeated addition?

 

[vanhees71]

The answer to such "why questions" is always either quite trivial ("Because it works!") or a deeply philosophical unanswerable question ("The incomprehensible effectiveness of math in the natural sciences", Wigner). Note, however that there's also an "incomprehensible ineffectiveness of philosophy in the natural sciences" (Weinberg).

The natural sciences wisely do not ask "why questions" but aims at an as accurate observation of natural phenomena possible (experimental physicists) and an as concise ordering of these empirical findings into ever simpler mathematical models. The amazing fact is that this works, i.e., that we find that we can sort all the empirical facts about the behavior of nature in an astonishingly simple way of a few basic principles, formulated in terms of symmetries of natural laws. The prize to pay is an ever higher level of abstraction, i.e., on the most fundamental level we have to describe the world in the quantum-theoretical formalism of operators, quantu fields, on a Hilbert space with the mathematical structure of the operators mostly determined by the symmetry principles underlying our (still classical, i.e., non-quantum description) of space and time or rather spacetime.

Why this works? Don't ask a scientist (see above ;-))!

 

[diegogarcia]

i notice with physics, all the equations are a bunch of scrambled sentences or variables being multiplied together. why the multiplication for everything?

Because, perhaps not surprisingly, multiplication and addition (and their inverses of division and subtraction) are the only operations that are unequivocal and absolute.

Physics, like all science, is based on measurement, and we measure things in terms of the the real number continuum. The only way to deal with this continuum is via multiplication and addition.

For example, we define units of measurement, such as the meter or the kilogram. But when we say that something has a mass of 2.5 kg this means: 1kg + 1kg + 0.5 kg. The multiplication step is needed because we need to proportion our mass unit with ratios (1/2) which involves multiplication.

Virtually all other operations in physics, and all science, are based on *, + and their inverses.

Derivatives are based on division, which is inverse multiplication.

Integrals, which are just a special kind of sum, are based on a combination of addition and multiplication.

Logarithms are defined by an integral (i.e. addition), and the inverse logarithm is exponentiation.

The trigonometric functions are based on integrals, or more precisely the inverse trig functions are.

The plethora of special functions, such as elliptic integrals, are again defined by integrals.

So we see than only addition and multiplication, which allow us to measure with a number continuum, are really the only operations that are unequivocal and absolute. Virtually every other operation in physics can be traced back to these fundamentals in some way.

I suppose that the human mind cannot conceive of anything beyond addition and the ratios that are expressed by multiplication.

 

[vanhees71]

I suppose that the human mind cannot conceive of anything beyond addition and the ratios that are expressed by multiplication.

That's obviously not true since the human mind has invented a lot more math than just these structures of algebra.

 

[diegogarcia]

That's obviously not true since the human mind has invented a lot more math than just these structures of algebra.

Could you provide some examples?

Also, keep in mind that the context of this discussion is physics. Addition and multiplication are operations that allow us to express measurements. They are not "structures of algebra."

What do we measure?

In mechanics we measure mass, position, and time, and we measure them as proportions (ratios) and/or summations of a basic unit that is defined with a real-number continuum. Everything else in mechanics, even energy, is a derived concept. (Some may argue that Lagrange/Hamilton begin with energy but this is just another, equivalent, way of doing things.)

The other types of things that we measure in physics can be ascertained by examining the fundamental units, such as temperature, charge, etc. Again, we measure using addition/multiplication and then build higher concepts based on these measures.

But my point is that addition and multiplication (and their inverses) are rock bottom. Everything other operation is merely an extension of them.

 

[PeroK]

But my point is that addition and multiplication (and their inverses) are rock bottom. Everything other operation is merely an extension of them.

There's Euclid's geometry to begin with.

 

[symbolipoint]

This is what. you spend your valuable time thinking about?

I do not know which emoticon to choose to show a reaction. My reaction I wish to pick is not an unfavorable one.

 

[diegogarcia]

There's Euclid's geometry to begin with.

Let's not drift backward from the context of MODERN physics and mathematics.

Modern science is based on the analytical methods that began with Descarte and which introduced the idea of a number continuum.

The ancients, Greek or otherwise, had no such idea. Their theoretical mathematics was based entirely on angles. (A lot more can be said but this simple statement is quite accurate.)

The ancients derived trigonometry, for example, from the ratios of angles within triangles. This approach may still be taught in high-school classrooms but the modern approach is to derive the trig functions analytically using integrals.

I'm not much of a Euclidian-style geometer but I believe that most compass-and-ruler constructions, for example, have analytical algebraic equivalents. This was how "squaring the circle" was finally proved to be impossible.

So any allusion to Euclid is really highly inappropriate. We must stick with modern ideas.

 

[PeroK]

Let's not drift backward from the context of MODERN physics and mathematics.

Riemannian manifolds and differential geometry are the modern ideas that built ultimately on the basics of axiomatic mathematics developed by Euclid.

Modern science is based on the analytical methods that began with Descarte and which introduced the idea of a number continuum.

The ancients, Greek or otherwise, had no such idea. Their theoretical mathematics was based entirely on angles. (A lot more can be said but this simple statement is quite accurate.)

The ancients derived trigonometry, for example, from the ratios of angles within triangles. This approach may still be taught in high-school classrooms but the modern approach is to derive the trig functions analytically using integrals.

I'm not much of a Euclidian-style geometer but I believe that most compass-and-ruler constructions, for example, have analytical algebraic equivalents. This was how "squaring the circle" was finally proved to be impossible.

So any allusion to Euclid is really highly inappropriate. We must stick with modern ideas.

Your original point was the limitations of the human mind. You're now, sadly, engaged in defending your hasty and ill-conceived idea by desperate arguments. Is it really so difficult to admit you were wrong?

 

[diegogarcia]

Lets not drift backwards from the context of MODERN physics and mathematics.

The ancient Greeks based their theoretical mathematics on angles. They had no concept of a number continuum. The trig functions, which the Greeks certainly understood, were derived from angle ratios.

Modern ANALYTICAL methods began with Descarte and the introduction of the number continuum. Trig functions are now derived analytically from integrals (although they still may be taught in schools using angles.

I am not much of a Euclid-style geometer, but I believe that most compass-and-ruler constructions, for example, have analytic algebraic equivalents. This was how the "squaring the circle" problem was finally proved to be impossible.

Drifting backwards to ancient times is really highly inappropriate.

 

[fresh_42]

This thread has served its purpose.

Inappropriate is definitely to tell others what is inappropriate! Highly! If at all it is the mentor's business to decide what is inappropriate.