Why did Galileo use multiplication instead of addition in his gravity equation?

[D H]

It's best to give up that notion that multiplication is repeated addition when you learn about fractions in elementary school. Eventually you'll come across mathematical structures where it makes no sense to think of multiplication as repeated addition. I would argue that the point this starts happening is the rationals. By the time you get to complex numbers, it's game over. Instead think of addition and multiplication as two distinct operations. That's what distinguishes a group from a ring.

Regarding the question of "Why not F=m+a", that doesn't make a bit of sense. You cannot add incompatible quantities. Physics is more than just numbers. Those numbers have units.

Regarding why not F=ma², or any formulation of the relation between force, mass, and displacement other than F=ma, that's not how the universe works.

Regarding the question of the minus sign, that's simply a matter of convention: Which direction, up or down, is positive, which is negative? Typically it's upwards that is designated as positive. Since gravitation is a downward force, with this convention it's F=-mg. Drop a rock down a well, however, and use depth rather than height as positive and it becomes F=mg.

 

[Dale]

But does it make any more sense to multiply the length of a carpet '2 metres times'

No, it does not make sense. It doesn't matter what the unit is, m, kg, ducks, or apples, the number of times an operation is applied is dimensionless (and a non-negative integer) and it simply makes no sense to do it a dimensionful number of times. Multiplication simply cannot be thought of as doing addition a certain number of times in physics.

 

[sophiecentaur]

It makes perfect sense that area (how much paint or carpet you need) is a length times a length. Integers or 'Reals' can be used and the operation is commutative, too. So 10 tiles of area pi involves the same operation as e times pi. The 'meanings' may not be the same but Maths is full of this sort of thing. We 'believe' the results of integrations and convolutions etc. So why pick on Multiplication to start non-believing? Familiarity breeds contempt, perhaps?

 

[Dale]

It makes perfect sense that area (how much paint or carpet you need) is a length times a length.

Of course it makes perfect sense to multiply a length by a length to get an area.

It makes sense precisely because multiplication is NOT the same thing as adding something to itself a certain number of times. If you add a length to itself an arbitrary number of times you always end up with a length, not an area. So clearly the operation of multiplying two lengths (which makes sense) is not the same as repeated addition of one of the lengths to itself the other length number of times (which doesn't make sense).

We 'believe' the results of integrations and convolutions etc. So why pick on Multiplication to start non-believing? Familiarity breeds contempt, perhaps?

I don't know what you are talking about here. Who is not believeing in multiplication? Who is contemptuous of multiplication? If you are referring to me then what have I said that indicates either of those? There must be a miscommunication somewhere.

 

[Naveen3456]

I want to say that I am a simple person and don't want to hurt anybody or prove anybody wrong.

I just want to clear my 'misunderstanding' of some well-established basic concepts. I also believe that in the pursuit of truth, we should throw our emotions/feelings out of the window.

I believe that math depicts nature/reality and therefore any equation that is correct should not give any answer that is in variance to reality/nature.

Let's consider F=ma.

Put a=1, and we get F=m


1. I fail to understand, how force can be equal to mass when both of them are altogether different things/concepts.



If you say, its not f=m, but it is the numerical values that are equal,

2. I fail to understand, how numerical values of unrelated things can ever become equal.




3. To my fragile mind, F=m only if force converts into mass and mass converts into force. But this is not the reality. So, why this thing is being shown by the equation.




If you say, F=ma depicts one interaction and F=m should not be taken seriously,

4. I think of E=mc2, where if c=1, E=m and this if indeed true. So, why F=m cannot be true and if it is not, why is it so in the equation. And why, F=m should not be taken on its face value.


I take it to be the 'failure' of my fragile brain that I am forced to ask such stupid questions.
Thanks.

 

[Dale]

Let's consider F=ma.

Put a=1, and we get F=m

You can never have a=1. You could have a=1g or a=1m/s^2 or a=1ft/min^2, but never a=1.

1. I fail to understand, how force can be equal to mass when both of them are altogether different things/concepts.

You are right, they always have different units.

If you say, its not f=m, but it is the numerical values that are equal[/I],

2. I fail to understand, how numerical values of unrelated things can ever become equal.

On the contrary, the numerical values depend on the choice of units. They can always be made equal through choice of units.

3. To my fragile mind, F=m only if force converts into mass and mass converts into force. But this is not the reality. So, why this thing is being shown by the equation.

It isn't. See above.

If you say, F=ma depicts one interaction and F=m should not be taken seriously,

4. I think of E=mc2, where if c=1, E=m and this if indeed true. So, why F=m cannot be true and if it is not, why is it so in the equation. And why, F=m should not be taken on its face value.

First, it is generally understood that c still has units of length over time, so it is merely a notational convenience. You are correct that it is technically an incorrect abuse of notation

Second, one very critical difference is that c is a constant and a is a variable. So you cannot generally set a=1 (in some units) through choice of units, e.g. If a varies during the experiment, but you can always set c=1 (in some units) through choice of units.

 

[sophiecentaur]

Of course it makes perfect sense to multiply a length by a length to get an area.

It makes sense precisely because multiplication is NOT the same thing as adding something to itself a certain number of times. If you add a length to itself an arbitrary number of times you always end up with a length, not an area. So clearly the operation of multiplying two lengths (which makes sense) is not the same as repeated addition of one of the lengths to itself the other length number of times (which doesn't make sense).

I don't know what you are talking about here. Who is not believeing in multiplication? Who is contemptuous of multiplication? If you are referring to me then what have I said that indicates either of those? There must be a miscommunication somewhere.

There is a lot more to this than you imply. All practical multiplication is basically integer arithmetic. We cannot multiply irrational or transcendental numbers. We always assume that the result of multiplying by pi will be 'somewhere between' one decimal number with a given number of places and the next one. What we do is to multiply by an integer number of 1/10000000000 ths. We always assume that a=bXc ( Algebra) works for all values but that's a matter of faith in Monotonicity, a Continuum, linearity and all the other facets of Analysis.
Maths is just a model - which happens to work well when used in a well behaved way but you can't take anything for granted. All we know is that we haven't actually found 'granularity' or extra dimension (as in string theory) in real life.

The essential thing when using Maths is good behaviour with Units. So a length times a length has units of length squared. However you do the multiplication (and that's what this thread is basically discussing) the numerical answer must carry the resulting units. Numbers on their own have no meaning in Science except when they are ratios (when the units cancel).

 

[Delta Kilo]

Many/most school-grade physics formulas are actually special cases of more general laws expressed in vector/tensor forms, where multiplication is replaced by some kind of 'product' operation, for example:
Newton's law $\textbf{F}=m\textbf{a}$ (scalar multiplication)
Mechanical Work $W = \textbf{F} \cdot \textbf{d}$ (dot product)
Angular momentum $\textbf{L} = \textbf{r} \times \textbf{p}$ (cross product)
Angular momentum $\textbf{L} =\textit{I}\textbf{w}$ (tensor product)
etc.

"Multiplication as a repeated addition" rule follows from linearity $(a+b)c = ac + bc$ and the existence of a unity element '1' such that $1a=a$. While different kinds of products are typically linear, they do not necessarily have unity element, so the rule does not apply to them.