Is there anything more to forces than being mathematical machinery?

[etotheipi]

Real forces arise between pairs of interacting bodies. For any given system, classical physics tells you how to compute the forces (i.e. Coulomb/Newton inverse square laws etc.) acting one body due to another body, and then also tells you how different parts of the system move under the action of those forces. It doesn't really tell you what the forces are though.

Really then, it appears that introducing the concept of forces is just a means to an end (i.e. you formulate your laws of physics using the construction of 'forces' and can start calculating results), since we could cut them out and just say that the interaction between particles causes motion (though admittedly it would be rather more complicated to calculate anything...). This might well be a stupid question, but is there a physical underlying motivation for introducing forces, or are they just a mathematical abstraction of the more fundamental interactions?

I'm just asking because I haven't been able to come up with a good description of what a 'force' is without referring to the definitions given the self-contained model of mechanics.

I apologise if this question has already come up before, it's one of those slightly "wishy-washy" ones that I'm sure you get a lot of but I couldn't find a satisfactory explanation for it!

 

[Dale]

It doesn't really tell you what the forces are though.

Why not? What would “really” tell you what they are? What does it mean to “really” tell what some X is?

I haven't been able to come up with a good description of what a 'force' is without referring to the definitions

To me, this is kind of an obvious thing. The definition of X is how we know what X is, so how could you expect to know what X is if you deliberately ignore the thing that tells you?

 

[etotheipi]

Why not? What would “really” tell you what they are? What does it mean to “really” tell what some X is?

If a particle is moving in space, I can measure its location with a set of orthogonal meter sticks for axes and build a giant arrow that I call its position vector. I can measure it's velocity (approximately) by waiting another second and then building another giant arrow, and finally by subtracting the arrows. Acceleration is the same. With lots of vector quantities, it's easy to visualise what the quantity is.

With 'force', all I can say is that I take my acceleration arrow and enlarge it by however much mass the thing has. I suppose this counts as a definition, but it's not very intuitive. It's not easy for me to pinpoint exactly what the force is, apart from that it is this multiple of the acceleration that arises because of some other thing I've put next to the particle.

Does that make any sense? It might not, I can't describe it too well (which is probably not a good sign...!).

 

[Dale]

If a particle is moving in space, I can measure its location with a set of orthogonal meter sticks for axes and build a giant arrow that I call its position vector. I can measure it's velocity (approximately) by waiting another second and then building another giant arrow, and finally by subtracting the arrows. Acceleration is the same. With lots of vector quantities, it's easy to visualise what the quantity is.

All of this can be said about force. You can perform measurements to determine what the force is just as you can perform measurements for the other quantities you mention. There is no difference whatsoever.

It's not easy for me to pinpoint exactly what the force is

That is probably because you are ignoring the definition. The definition of X is the thing that tells you what X is. How can you expect to know what something is “without referring to the definition“?

 

[PeroK]

At a fundamental level you might start to ask what space and time really are, but the question about forces is more to the point. Newton was well aware that although his law of gravitation explained the motion of the planets, for example, there was no rational mechanism for this force:

It is inconceivable that inanimate Matter should, without the Mediation of something else, which is not material, operate upon, and affect other matter without mutual Contact…That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it. Gravity must be caused by an Agent acting constantly according to certain laws; but whether this Agent be material or immaterial, I have left to the Consideration of my readers.[5]
— Isaac Newton, Letters to Bentley, 1692/3

The electromagnetic force in the classical theory has the same issue. If we say that all classical forces are either gravitational or electromagnetic, then quite explicity we know how to measure a force and how to model it, but the theories say nothing about how that force is transmitted. Or "what it really is".

 

[Mark44]

This might well be a stupid question, but is there a physical underlying motivation for introducing forces, or are they just a mathematical abstraction of the more fundamental interactions?

Maybe I'm not understanding your dilemma, but one underlying motivator for the concept of force is the relative difficulty of lifting an object and moving it to some other place. One factor would be the object's weight. For example, my house is situated near a stream. About 14,000 years ago, my property was covered by a layer of ice maybe 2,000 ft. thick. As the ice moved downstream, it carried rocks of all sizes with it, but when the ice melted away, the rocks were left in place. As a result, there are rocks of all sizes here an there on my property. Most of these are of a size that I can lift, but I've found one that was too heavy for me to lift, and there's another huge boulder that would be too heavy for anything but a high-capacity crane to lift.

My point is that the weight of an object, the force of attraction between the Earth and the object, is anything but a mere mathematical abstraction.

 

[FactChecker]

I think that you are asking if the abstraction of "force" is essential. The abstraction allows it to be used in many situations that would not fit anyone specific non-abstract definition.

 

[etotheipi]

...I have left to the Consideration of my readers

Newton falling for the old "I have left [insert impossibly difficult thing] as an exercise to the reader" trap...

If we say that all classical forces are either gravitational or electromagnetic, then quite explicity we know how to measure a force and how to model it, but the theories say nothing about how that force is transmitted. Or "what it really is".

That's fair enough, it's sort of what I'd presumed was the resolution. Just a directed line segment associated with the interaction of one body on another in such a way that you get $\mathbf{F} = m\mathbf{a}$, in classical physics anyway. The interaction is real, the force is just a way of modelling it, which I guess makes it just as real too.

I know essentially nothing about relativistic theories since I have barely studied them, but I am aware that in that arena you get $\mathbf{F} = \gamma^{3} m\mathbf{a}$. I presume in that definition it is the acceleration that changes and not the force (i.e. all that strange coordinate/proper stuff). Muddies the water a little bit in terms of defining what a force is, but I don't really want to go down that road since I don't know enough for it to be meaningful...

 

[FactChecker]

Just a directed line segment associated with the interaction of one body on another in such a way that you get $\mathbf{F} = m\mathbf{a}$, in classical physics anyway.

But even there, what about the situation where there are two opposing "forces" giving no actual acceleration? $F=mA$ works as a definition of net force, but there is a lot of use of forces that are opposed and result in no acceleration.

 

[etotheipi]

But even there, what about the situation where there are two opposing "forces" giving no actual acceleration? $F=mA$ works as a definition of net force, but there is a lot of use of forces that are opposed and result in no acceleration.

Yeah, I guess the additional stipulation that it's the only force for the purposes of the definition is needed.

Thanks for your replies, I just thought it was slightly curious. At the end of the day classical physics is just a model that happens to be fairly consistent with observation and the nature of the forces themselves don't really matter, just so long as they're consistent with the model. I suppose you could say the same thing about energy or indeed any physical quantity, as @Dale alluded to.

 

[Dale]

If we say that all classical forces are either gravitational or electromagnetic, then quite explicity we know how to measure a force and how to model it, but the theories say nothing about how that force is transmitted. Or "what it really is".

I disagree. When a theory defines X that is the theory explicitly saying what X is.

I categorically dislike questions containing the word “really”. If you ask “what is force” people simply respond (correctly) with the definition. Ask “what is force really” and suddenly everyone is afraid to respond (correctly) with the definition. The word “really” does not change the answer.

 

[PeroK]

I disagree. When a theory defines X that is the theory explicitly saying what X is.

I like that quotation from Newton because it shows that he knew there must be something more fundamental going on. The popular picture is often of this great man who thought he had explained everything and who would have been horrified by GR. However, given what he said about his own theory, if you brought him back in 1915 and showed him GR I imagine he would have said "I knew there was more to it"!

 

[etotheipi]

I categorically dislike questions containing the word “really”. If you ask “what is force” people simply respond (correctly) with the definition. Ask “what is force really” and suddenly everyone is afraid to respond (correctly) with the definition. The word “really” does not change the answer.

I do apologise, that wasn't the intent of my OP! If the best answer is that force is a quantity defined as $m\mathbf{a}$, I would consider that a perfectly good answer. Since in the context of the OP, the conclusion would be that, yes, force is a mathematical construction (i.e. it arises from a precise mathematical definition). If instead it were the case that force had another underlying interpretation that I wasn't aware of, and that it was solely modelled by $m\mathbf{a}$ in classical mechanics, that would contradict the title and be another answer.

I've no issues with the definitions within the confines of Newtonian mechanics, was just interested to see if anyone knew something cool that I was unaware of .

 

[Vanadium 50]

I categorically dislike questions containing the word “really”.

Me too.

Ask “what is force really” and suddenly everyone is afraid to respond (correctly) with the definition. The word “really” does not change the answer.

Ah, but what is it really really?

The thing i don't like the word "really" is that it allows the OP of a thread to think he is asking a clear question when in fact he or she is not. The subtext is usually "I know the definition, but am dissatisfied with it in some way", which is fine, but by allowing the "really" construct, it leaves the responders guessing as to what the concern really is.

 

[kuruman]

I can't claim to have read Newton's mind, but I like to think that he invented forces as part of a method for understanding the universe. What happened is probably something like this. Newton looked at the universe around him and thought, "I can't understand this all at once. I need to chop it up into little pieces, study each piece separately, reassemble the pieces and with some luck and educated guesses I will reach new insights about how the pieces work together."

Of course when studying each piece (now called "the system") you cannot really separate it from the others because they affect it. So if you want to ignore the rest of the universe and concentrate on the system, you have to replace the effects of the universe on the system with something that mimics these effects. That particular something is a "force", sometimes exemplified by the disembodied hand often seen in physics textbooks. That kind of reasoning was, I think, the motivation for the genesis of the Newtonian idea of force. The first and third laws follow directly from this idea. First law: "If you leave it alone, it will keep on doing what it has always been doing". Thirdlaw : "If you let it know that you are there, it let's you know that it is there." The second law is just a quantification of the effect that the sum of all the forces has on a system.

So to me, a force is a mode of interaction of a system with the rest of the universe. If the universe exerts no forces (not zero net force) on the system, the system does not know that the universe exists and neither does the universe know that the system exists. We deduce the existence of forces by examining the motion of objects after having postulated that objects change their state of motion because of forces.

 

[etotheipi]

The thing i don't like the word "really" is that it allows the OP of a thread to think he is asking a clear question when in fact he or she is not. The subtext is usually "I know the definition, but am dissatisfied with it in some way", which is fine, but by allowing the "really" construct, it leaves the responders guessing as to what the concern really is.

I was hesitant to start the thread precisely for this reason, I couldn't really articulate it well at all which makes it hard for you guys to help out. So I am sorry, I'll try to steer clear of open-ended questions like this in the future.

That said, all of the replies are genuinely helpful so thank you everyone for putting up with me !

 

[wrobel]

I'm just asking because I haven't been able to come up with a good description of what a 'force' is without referring to the definitions given the self-contained model of mechanics.

There are only experiments and mathematical models that describe\predict them. No metaphysical entities of forces or any other physical constructions exist

 

[DrStupid]

If the best answer is that force is a quantity defined as $m\mathbf{a}$, I would consider that a perfectly good answer.

Force is defined as something that needs to be impressed upon a body to change its state of rest or uniform motion. Newton never specified the nature of force - e.g. if it is "material or immaterial" (see citation in #5). There was no reson to do so. That would just raise philosophical discussions without any benefit.

The original second law is btw. not F=m·a but F=dp/dt. That still holds in special relativity.

 

[Herman Trivilino]

At the end of the day classical physics is just a model that happens to be fairly consistent with observation and the nature of the forces themselves don't really matter, just so long as they're consistent with the model.

The thing is, though, scientists, engineers, and technicians can use the concept to do stuff and to build stuff. That is where the value really lies, in the utility of the concept.

 

[apollo]

For a total layman's perspective with grade 8 mathematics, isn't force 'really' just how much something pushes on something? Or pushes back, if stationary?

 

[FactChecker]

You should probably say that force is F=mA or that force is the ability to cause acceleration of a mass. That gives it units and a way to specify the amount even if there is an opposing force that stops any acceleration.

 

[apollo]

The ability to cause acceleration of another mass or of the mass in question who's force we are measuring?

 

[jbriggs444]

The ability to cause acceleration of another mass or of the mass in question who's force we are measuring?

We evaluate a force based on the acceleration of the thing that it acts on.$$\sum F=ma$$The $\sum F$ is the sum of all forces that act on the object whose mass is $m$. The acceleration of that object will be $a$.

If we are talking about a pair of objects pushing or pulling on one another, Newton's third law tells us that the force of the one on the other will be equal to that of the other on the one. So it does not matter much which object we use to determine how much force there is. Use the one that makes for the easiest measurement.

 

[Herman Trivilino]

For a total layman's perspective with grade 8 mathematics, isn't force 'really' just how much something pushes on something? Or pushes back, if stationary?

There's no need for the "if stationary" caveat.

I don't think there's any such thing as a total layman's perspective. There's a degree to which any concept can be understood, and sometimes it can go very shallow, but only if it leaves out something that might be important..

 

[FactChecker]

There's no need for the "if stationary" caveat.

I think the point is that the absence of acceleration does not always imply the absence of any force. It would be nice if the definition of force did not require an actual acceleration.
The definition of force does not really require an actual acceleration:
From https://en.wikipedia.org/wiki/Force: "a force is any interaction that, when unopposed, will change the motion of an object. A force can cause an object with mass to change its velocity (which includes to begin moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of Newtons and represented by the symbol F. "

 

[DrStupid]

It would be nice if the definition of force did not require an actual acceleration.

It doesn't. The definition of forces says that they are proportioanl (or equal if you use proper units) to change of momentum. If you have a force acting on a body it changes it's momentum by $\dot p$. If there is another force that changes its momentum by $-\dot p$ than there is no acceleration, even though both forces comply with Newton II and would result in an acceleration if they would act alone. You can also calculate the individual accelerations and add them to get the total acceleration of the body. However, I prefer adding the forces in order to get the acceleration as described by jbriggs444 in #23 above.

 

[etotheipi]

If you have a force acting on a body it changes it's momentum by $\dot p$. If there is another force that changes its momentum by $-\dot p$ than there is no acceleration,

That's actually quite an interesting take on it, since I've only really ever considered the $\dot{p}$ due to the resultant force. I suppose you could think of it in terms of each force pumping in or siphoning out momentum, and just adding together all of the changes.

 

[archaic]

Relevant to the topic
https://physics.stackexchange.com/q...-motion-laws-or-definitions-of-force-and-mass
https://physics.stackexchange.com/questions/340829/definition-of-mass-in-Newtonian-mechanics

 

[FactChecker]

It doesn't. The definition of forces says that they are proportioanl (or equal if you use proper units) to change of momentum. If you have a force acting on a body it changes it's momentum by $\dot p$.

The problem with requiring any actual change of momentum is the same as the problem with requiring an actual acceleration. That does not always happen.

If there is another force that changes its momentum by $-\dot p$ than there is no acceleration,

If there is no change of momentum then your definition gives no force of any kind.

Your original statement "Force is defined as something that needs to be impressed upon a body to change its state of rest or uniform motion. " is one that, without explicitly stating it, does not require that there actually be a "change of state of rest or uniform motion", just that it "needs to be impressed" to make such a change. That is a distinction which only obsessive/compulsive people might care about. (I plead guilty.)

 

[etotheipi]

I think @DrStupid's suggestion was taking a principle of superposition approach; i.e. if $\vec{F}_1 = \dot{p}_1$, $\vec{F}_2 = \dot{p}_2$ when acting individually then $\vec{F}_1 + \vec{F}_2 = m(\dot{v}_1 + \dot{v}_2) = \dot{p}_t$. Since the accelerations and rate of change of momenta are linear in force, it's valid to consider each action individually and then superpose them at the end.

Whilst of course the physical law is that the resultant force is $\frac{d\vec{p}}{dt}$.

 

[FactChecker]

I think @DrStupid's suggestion was taking a principle of superposition approach; i.e. if $\vec{F}_1 = \dot{p}_1$, $\vec{F}_2 = \dot{p}_2$ when acting individually then $\vec{F}_1 + \vec{F}_2 = m(\dot{v}_1 + \dot{v}_2) = \dot{p}_t$. Since the accelerations and rate of change of momenta are linear in force, it's valid to consider each action individually and then superpose them at the end.

Whilst of course the physical law is that the resultant force is $\frac{d\vec{p}}{dt}$.

Yes, I agree. I am afraid that I was getting obsessed with a detail and should leave it alone.

 

[etotheipi]

Yes, I agree. I am afraid that I was getting obsessed with a detail and should leave it alone.

No I think your objection was quite valid, since there wasn't an explicit mention of superposing solutions. I also agree that I find the definition of forces slightly unsatisfying, but it appears to be the case that we can't do much better .

 

[DrStupid]

I suppose you could think of it in terms of each force pumping in or siphoning out momentum, and just adding together all of the changes.

That's exactly how I see it. Force is an exchange of momentum between two systems. That implies both Newton II and Newton III. The energy equivalent would be the power of work and heat. According to the first law of thermodynamics the sum of work and heat transferred to a system is equal to the change of its internal energy. But that does not mean that there is no work or heat if the internal energy remains constant.

 

[DrStupid]

Your original statement "Force is defined as something that needs to be impressed upon a body to change its state of rest or uniform motion. " is one that, without explicitly stating it, does not require that there actually be a "change of state of rest or uniform motion", just that it "needs to be impressed" to make such a change.

I am quite sure that it is ment like that because Newton repeated it in the first law of motion. A force is required to change the state of motion but it is not sufficient. Several forces and the corresponding changes of momentum (and therefore also the resulting accelerations) can add to zero.

 

[etotheipi]

That's exactly how I see it. Force is an exchange of momentum between two systems. That implies both Newton II and Newton III. The energy equivalent would be the power of work and heat. According to the first law of thermodynamics the sum of work and heat transferred to a system is equal to the change of its internal energy. But that does not mean that there is no work or heat if the internal energy remains constant.

That's certainly a nice way of thinking of it!

 

[Dale]

The definition of forces says that they are proportioanl (or equal if you use proper units) to change of momentum.

That is pretty close to my preferred definition: “force is the rate of transfer of momentum”. Since momentum is a conserved quantity it’s makes sense to speak of transferring it. Both Newton’s 2nd and 3rd laws follow directly from that definition.

Edit: I see you said as much in a subsequent post!

 

[archaic]

That is pretty close to my preferred definition: “force is the rate of transfer of momentum”. Since momentum is a conserved quantity it’s makes sense to speak of transferring it. Both Newton’s 2nd and 3rd laws follow directly from that definition.

Edit: I see you said as much in a subsequent post!

Just for curiosity, how do you define mass? Something like volume times density would be tautological.

 

[jbriggs444]

Just for curiosity, how do you define mass? Something like volume times density would be tautological.

Historically, it had been defined in terms of another standard mass, using things like balance scales -- forces and torques as proxies for mass under the assumption of a locally uniform gravitational field. [One can also use spring compression as a proxy for force and make assumptions about temporally uniform gravitational fields and linear and unchanging spring properties].

The idea being that two bricks have twice as much mass as one. Or that a lump of platinum irridium over here that balances with a lump of platinum irridium over there must have the same mass.

The underlying assumptions about linearity, uniformity, and such have been well tested for millenia. Commerce and commercial advantage is a powerful motivator.

 

[Dale]

Just for curiosity, how do you define mass? Something like volume times density would be tautological.

I would define mass operationally. Something like “mass is the quantity measured by a balance scale”. This is similar to how time is defined.

 

[kuruman]

I would define mass operationally. Something like “mass is the quantity measured by a balance scale”. This is similar to how time is defined.

In that case wouldn't you have to define a uniform gravitational field or equivalent since the balance scale doesn't work very well in free space?

 

[FactChecker]

I am quite sure that it is ment like that because Newton repeated it in the first law of motion. A force is required to change the state of motion but it is not sufficient. Several forces and the corresponding changes of momentum (and therefore also the resulting accelerations) can add to zero.

Yes. The way you stated it there makes sense and takes care of my concern.

 

[DrStupid]

Both Newton’s 2nd and 3rd laws follow directly from that definition.

Only if you do not insist on a limitation of Newton III to two-body forces because that does not follow directly from the interpretation of force as transfer of momentum .

 

[DrStupid]

Just for curiosity, how do you define mass?

In classical mechanics it is indirectly given by the definition of momentum, Newton II + III, the principle of relativity, isotropy and the transformation. (But be careful: If you use this definition in relativity you get something that is like a red rag to a bull for physicists.) I do not know if it also works without isotropy. It is at least required in the one-dimensional case.

Something like volume times density would be tautological.

Is is a tautology today because we use this definition the other way around to define density. I do not think that was the case when Newton defined mass this way centuries ago. It provides at least a general impression what mass is and what it not is (e.g. amount of substance, which is sometimes confused with Newtons term "quantity of matter").

 

[Dale]

In that case wouldn't you have to define a uniform gravitational field or equivalent since the balance scale doesn't work very well in free space?

I would leave details like that for the manufacturer of the balance scale to explain in the owner’s manual. I would not make it part of the definition.

 

[Dale]

Only if you do not insist on a limitation of Newton III to two-body forces because that does not follow directly from the interpretation of force as transfer of momentum .

I still strongly disagree with your take on that topic but have less than 0 interest in renewing a discussion on it.

 

[jbriggs444]

I would leave details like that for the manufacturer of the balance scale to explain in the owner’s manual. I would not make it part of the definition.

Lack of gravity is not a problem encountered in typical use but artificial gravity (e.g. rotation) can readily be employed if needed. The Acme balance scale, orbital version, has the requisite information in appendix B.

 

[sophiecentaur]

It doesn't really tell you what the forces are though.

I don't think this is anything more than a way of communicating a shared experience with other people - same as any other piece of language. You can draw up a list of effects that can be attributed to the result of a "force'. The introduction of Maths into it makes no fundamental difference; Maths just improves the precision of communication between people and allows more reliable 'reasoning'.
You and I (I'm assuming) had similar educations so we can communicate about a process that involves what we both call "a force" by using Newton's Laws of Motion. If you try to build up knowledge about the World without using Science, you end up with statements like "Nature abhors a vacuum" and we know that seriously falls short when you want to build a rocket or a radio set.
Otoh, if you want to get all philosophical then you need to go elsewhere than PF for your answers.

 

[atyy]

I do apologise, that wasn't the intent of my OP! If the best answer is that force is a quantity defined as $m\mathbf{a}$, I would consider that a perfectly good answer.

That is only part of the answer. Consider $F = ma$. This is not that physically meaningful until we say what $F$ is. $F$ is a push or a pull - an interaction - between two (or more) bodies, with the strength and direction of the push or pull being dependent on the properties of both bodies.

In the case, of gravitation, the relevant properties of the bodies are their masses: $F = Gm_{1}m_{2}/r^{2}$

In the case of the electrical force (in electrostatic situations), the relevant properties of the bodies are their charges: $F = Kq_{1}q_{2}/r^{2}$

Combining the electrostatic force law with Newton' second law gives $Kq_{1}q_{2}/r^{2} = ma$, which shows that the properties of bodies that determine how they interact (ie. the LHS of $F=ma$) do not necessarily involve the inertial mass (in the RHS of $F=ma$).

In the case of the gravitational force, the property on the LHS is called the gravitational mass. Amazingly, in the unique case of the gravitational force, a body's gravitational mass (on the LHS) is always proportional to its inertial mass (on the RHS). This coincidence is one form of something called the Principle of Equivalence, which inspired Einstein's approach to General Relativity, a relativistic theory of gravity. Reformulated versions of the Principle of Equivalence hold in General Relativity.

 

[atyy]

Really then, it appears that introducing the concept of forces is just a means to an end (i.e. you formulate your laws of physics using the construction of 'forces' and can start calculating results), since we could cut them out and just say that the interaction between particles causes motion (though admittedly it would be rather more complicated to calculate anything...). This might well be a stupid question, but is there a physical underlying motivation for introducing forces, or are they just a mathematical abstraction of the more fundamental interactions?

The concept of force is the way the interactions are described in elementary Newtonian physics. In relativity, and especially relativistic quantum mechanics, the force concept of Newton is not so useful. But although we no longer use the mathematical formalism of Newton in those cases, we still call the interactions between bodies "forces".

And yes, even classically you can get rid of forces eg. the Hamiltonian and Lagrangian formalisms.

 

[sophiecentaur]

And yes, even classically you can get rid of forces eg. the Hamiltonian and Lagrangian formalisms.

Good point BUT I very seldom come across a situation where my body tells me I'm dealing with Potential Energy, rather than with Forces.

I've had many useful conversations with young teenagers about the 'nature' of Forces and Energy and they clearly aren't ready for any more than what our experiences of those two things involves. To my mind, that implies that we start off with lists of what effects can be put down to Force or Energy. I have never been asked (at that level) what they 'really are'.

Many people want to fill in that gap between the totally intuitive / subjective experience and the mathematical model by looking for a 'really is' model. But, without the Maths, the 'really is' is little more than a dressed up version of intuition.