Why is the mathematical model favored over the mechanical model?

[wmikewells]

In theoretical physics, why is the mathematical model favored over the mechanical model?

Awhile back, I posted a thread asking about what each theory posits as real. For example, quantum field theory might limit the set of real things to fields, field quanta, the universe, and causality. It was an interesting exercise for me to see how each theory highlights different "real" things even if the proponents denied making any statements about what was real.

However, in almost all instances when I made an inquiry, the response is typically underwhelming. I often hear that it is not an important question, that it should be left up to philosophers, or that defining what is real is unnecessary to do physics. For example, I have often read and heard that spacetime does not correspond to anything real. It is a mathematical representation of what happens, and it is sufficient to explain why it happens. In other words, worrying about reality just muddles the picture.

All questions about existence seem to be relegated to confirming mathematical models. For example, confirming the existence of black holes and the Higgs Boson were milestone events. They confirmed the validity of the underlying math.

While I would agree that mathematical models are necessary and powerful tools, which can lead to new and important discoveries, I am curious as to why mechanical models are largely ignored. In other disciplines, they make up the discipline's bread and butter. For example, the DNA molecule is the foundation of modern genetics. So, I am left to wonder why theoretical physics is the exception.

Below is a list of all the reasons I could think of to explain why mechanical models are not used in theoretical physics. I tried my best to state each as a non-straw man argument. It is not my purpose to start a discussion about the merits of each (even though I have qualms about each which I would be willing to discuss off-line). Rather, I am just looking to create a list of arguments that has at one point or another been proposed or would be agreed to. Some of the arguments may overlap.

1. Any mechanical model can be represented mathematically, making the mechanical model irrelevant (Example: spacetime).
2. As a practical matter, theoretical physics just does not lend itself to mechanical modeling.
3. Positing real things in a mechanical model requires a definition of reality, which is outside the scope of physics.
4. Quantum mechanics tells us that reality is truly indeterminate until it is measured and therefore incapable of being mechanically modeled (Example: the separation of the micro and macro-world).
5. The uncertainty principle provides a boundary, beyond which mechanical models won't work.
6. Mechanical models are at best analogies (Example: stretched rubber sheet to represent gravity).
7. Mechanical models cannot be employed until we get a better understanding of all physical phenomenon (Example: unification of all forces).
8. Causality, which is required for mechanical models, is an emergent property, and therefore cannot be used to explain phenomenon at the micro-world level.
9. Mechanical models are an invention the human mind imposes on physical phenomenon to understand the world and therefore cannot be used to accurately model the true world (Example: the planet model of the atom where the electron orbits the nucleus).
10. The universe works according to laws and not mechanics.

 

[sophiecentaur]

Some good points there and it is a summary of 'why not'. Imo, the main reason not to use a mechanical model is that for it to be any use, it needs to have a quantitative correspondence with the actual situation. That is to say, you would need to be able to plug in values and to get useful answers out. This requirement leaves you needing to build an Analogue computer for each situation. (Springs, motors, friction material etc. etc. - and then how would you take this into QM?) Why bother, when you can just drop some numbers into a mathematical equation?
People who claim that the only way they could get into Science is with mechanical ("Physical") models are totally excluding themselves from most of the juicy stuff. They could never hope to bash any frontiers. It's a recipe for frustration - as the early Scientists found.

Basically, the Universe is just not 'Mechanical'.

 

[wmikewells]

Thanks for the reply. In thinking about it, the argument that the Universe is basically not mechanical could probably be used to derive most of the arguments I outline above. Would you mind expanding a little more on the differences you see between a mechanical and a mathematical universe?

For what it is worth, my two cents is that mathematical and mechanical modeling should go hand in hand. My prejudice-for-today is that mechanical models have seen limited action not because of a property of the universe itself but because the objects in our reality toy box (QM particles and fields) are inadequate to generate mechanical models. In other words, the lack of mechanical models is symptomatic of a problem with the QM objects themselves. Your reply has put me on a new path I have yet to explore, so I might have to think about it for a bit. It might be a dead-end.

 

[wmikewells]

I was on a roll, so I took a stab at differentiating between a mechanical universe and a mathematical universe. The comparison may be too simplistic, but I think it covers all the important differences. Again, the goal is not to generate a discussion over which universe is correct. A table would have been a better choice for the comparison, but I just used simple numbering.

Mechanical Universe
1. Only physical objects exists.
2. Physical objects interact to create events.
3. Events are related through causality to create reality.
4. Truth is dependent on reality.

Mathematical Universe
1. Only mathematical objects exist (for example, quantities, probabilities, sets, etc).
2. Mathematical objects are applied to phenomenon to create properties (for example, mass, polarization, etc.). Note: this requires rewording or revision. Something doesn't seem right.
3. Properties are related through laws to create truth.
4. Reality is dependent on truth.

Any feedback would be appreciated.

 

[sophiecentaur]

Neither of the two approaches is totally reliable. The mechanical approach falls short because it cannot include all phenomena that we have observed, so far. The mathematical approach is 'wider' than our reality so it can yield impossible solutions that take us outside our reality.

On the whole, the mathematical approach is more likely to get us somewhere. The mechanical approach is always going to let us down. However, the concrete approach is necessary for making a start and for getting across ideas in the form of metaphors.

I, personally, do not accept that there is an ultimate 'truth' - certainly not one that a intelligence, such as the human mind could grasp because we are a part of the reality we try to model. This would constitute a map of a map of a map . . . . ., which is not possible.

 

[Benit13]

For me, mechanical and mathematical approaches are the same thing and there is no need for a dichotomy:

Universe:
1. Physical entities can be perceived by us and mathematical formulations can be made/discovered that describe those objects. Whether they exist or not depends on your solipsist or realist point of view.
2. Physical systems, which are characterised by properties, change with time, giving rise to events. These physical systems follow mathematical rules that depend on the properties of the system and its surroundings.
3. We can determine common laws and properties to learn more about the laws of physics. This body of knowledge and facts is often referred to by us as "the truth". This differs from your idea of truth, where I think you are asserting that truth "is" the laws of nature. For me the laws of nature are a universal set of laws that are part of reality and truth is a human invention that is a slowly growing subset (for the most part at least...) of that universal set.
4. We decide what is real and what is truth based on our theories and observations. Considering point 3 it makes no sense to debate which is dependent on which. It is simple; physical entities can be completely characterised by what they are (at a given point in time) and how they behave over time as shown by physical laws.

Let's forget quantum weirdness for the moment :D

 

[sophiecentaur]

For me, mechanical and mathematical approaches are the same thing and there is no need for a dichotomy:

It all depends on the definition of "Mechanical'. The sort of concrete model that many PF contributors seem to want is certainly not adequate and usually indicates that they can't or won't do the Maths. If the mechanical model is maths based (by implication and with validation) then there is no dichotomy and it's the same thing - just with different use of variables.

To my mind, avoiding the use of Maths is like trying to describe an everyday process, using the English language but leaving out all the verbs. There is no language that can substitute for Maths for a lot of Science. If the conclusion from this thread is that Maths is not necessary then there will be many readers (possibly we have more readers than just the few of us contributors) who will feel justified in Mathemaphobia. That would be a shame. I speak as one for whom Maths has always been a struggle - but I still appreciate how essential it is.

 

[wmikewells]

Thanks for your thoughts. I presented the dichotomy to best state (without judgement) some of the positions I have seen floating around. I, like you, think that both approaches are needed.

For me, mechanical and mathematical approaches are the same thing and there is no need for a dichotomy:

I would agree that both approaches are similar. They both use the same tool set with minor differences. That is why they parallel each other. It was one of the things I discovered when outlining the approaches. However, I would maintain that there is a qualitative difference between the two. For the mechanical universe, objects are discovered to exist, while in the mathematical universe, they are assumed to exist.

Universe:
1. Physical entities can be perceived by us and mathematical formulations can be made/discovered that describe those objects. Whether they exist or not depends on your solipsist or realist point of view.

I would add that physical entities can also be inferred to exist, but I am guessing that you would agree to that. As for mathematical models that describe objects, I would argue that the question is irrelevant. Only the assumption of existence is necessary.

2. Physical systems, which are characterised by properties, change with time, giving rise to events. These physical systems follow mathematical rules that depend on the properties of the system and its surroundings.

You lost me here. Please elaborate.

3. We can determine common laws and properties to learn more about the laws of physics. This body of knowledge and facts is often referred to by us as "the truth". This differs from your idea of truth, where I think you are asserting that truth "is" the laws of nature. For me the laws of nature are a universal set of laws that are part of reality and truth is a human invention that is a slowly growing subset (for the most part at least...) of that universal set.

Again, I was trying to be the mouthpiece for a position I don't really support. I didn't really require any particular definition of truth, only that it takes precedence over reality, which is a position I don't support.

4. We decide what is real and what is truth based on our theories and observations. Considering point 3 it makes no sense to debate which is dependent on which. It is simple; physical entities can be completely characterised by what they are (at a given point in time) and how they behave over time as shown by physical laws.

I would agree with you. Separation of reality and truth creates problems. However, I think some have been forced to that separation from a conflict between certain assumptions about our universe and "quantum weirdness". See comment below.

Let's forget quantum weirdness for the moment :D

I think it is quantum weirdness that has caused many to abandon all hope for mechanical models. I would maintain that the next great revolution in physics will be the development of a mechanical model that explains quantum mechanics along with everything else. I think gravity and all the other forces could be unified just using mathematical models; however, I wouldn't be surprised if all attempts failed until the mechanical model was presented.

 

[wmikewells]

To my mind, avoiding the use of Maths is like trying to describe an everyday process, using the English language but leaving out all the verbs. There is no language that can substitute for Maths for a lot of Science. If the conclusion from this thread is that Maths is not necessary then there will be many readers (possibly we have more readers than just the few of us contributors) who will feel justified in Mathemaphobia. That would be a shame. I speak as one for whom Maths has always been a struggle - but I still appreciate how essential it is.

I too am mathematically challenged. I didn't have an issue when I was younger. I lost any gift I had as I got older.

It is rare to find someone who has both math and mechanical ability. It is almost as if a team is required to combine the two.

 

[sophiecentaur]

I too am mathematically challenged. I didn't have an issue when I was younger. I lost any gift I had as I got older.

It is rare to find someone who has both math and mechanical ability. It is almost as if a team is required to combine the two.

I mentioned before that we haven't actually decided a proper definition of "mechanical" and your statement demonstrates it. In many peoples' minds there is another Culture Split, between intellect and practical (mechanical?) ability. I agree that you can see it at work in many places. However, I worked with some very clever (brilliant, even) people in Engineering and I actually found that, at that level, the best of them were not only good at Maths but had excellent experimental skills. It used to interest me that many of these bright guys had very little interest in culture (except music). Shakespeare was "rubbish" for many of them.

 

[Vanadium 50]

This thread is - at best - philosophical.