Does Math always correspond to reality?

[ViolentCorpse]

Hello everyone!

I got into a debate with a friend about a month back and this question has been racing through my head ever since. Does Math necessarily reflect what's real?

He asserted that Math can give you results far-off from what's real. He illustrated his claim like this: Suppose you want to build an average-sized building. You employ some workers for the job and and then estimate mathematically the time the building should take to complete. You get a certain estimate but it occurs to you that if you employ more workers, it should take less time. Suppose then that if you add just enough workers, your calculation would give you a time estimate to complete the building of 5 minutes. But that is impossible in reality. No matter how many workers you add, you can not build and average-sized structure that fast!

There has to be something gravely wrong in his reasoning but I can't pinpoint it. I'm not good at anything and I actually feel embarrassed bringing this stupid argument up here to you people of such high understanding; but science and math are the only two things in the world I have my full confidence in and the idea that math can be so disjointed from reality is making me very uneasy.

I tried to keep it as short as possible to ensure that I do not consume too much of your precious time, but I guess I failed. I'm sorry about that.

So, what's your take on this?

Thank you so much for your time! ^_^

 

[phinds]

Math can always be misapplied to give silly results like that. That is not math's fault and is NOT any kind of indictment of math, just of the moron who applied it. Math is not the ONLY thing you have to take into account when applying math to real situations.

In a different sense than I think you asked, math does NOT always correspond to reality, not because it is misapplied but because it was constructed separate from reality.

In earliest human times, math was all "applied math", like counting things.

These days, there is abstract math that is not intended to correspond to reality (but of course that's NOT what's going on with your example), it is just intended to be self-contained and self-consistent without regard to anything outside of its own logical construct.

 

[ViolentCorpse]

Math can always be misapplied to give silly results like that. That is not math's fault and is NOT any kind of indictment of math, just of the moron who applied it. Math is not the ONLY thing you have to take into account when applying math to real situations.

In a different sense than I think you asked, math does NOT always correspond to reality, not because it is misapplied but because it was constructed separate from reality.

In earliest human times, math was all "applied math", like counting things.

These days, there is abstract math that is not intended to correspond to reality (but of course that's NOT what's going on with your example), it is just intended to be self-contained and self-consistent without regard to anything outside of its own logical construct.

Yeah, that's surprisingly what I thought too. I mean Maths is not sentient and it doesn't take into account every possible thing that can influence an event on its own. To use it effectively, you must be a good user first. But I think what he was getting at was that, theories in theoretical physics are based on mathematics and not on experiment (since he's not a believer in science). So how can we establish the truth of such theories? By testing the predictions it makes against the observations, I suppose?

 

[SteveL27]

Hello everyone!

I got into a debate with a friend about a month back and this question has been racing through my head ever since. Does Math necessarily reflect what's real?

He asserted that Math can give you results far-off from what's real. He illustrated his claim like this: Suppose you want to build an average-sized building. You employ some workers for the job and and then estimate mathematically the time the building should take to complete. You get a certain estimate but it occurs to you that if you employ more workers, it should take less time. Suppose then that if you add just enough workers, your calculation would give you a time estimate to complete the building of 5 minutes. But that is impossible in reality. No matter how many workers you add, you can not build and average-sized structure that fast!

You might enjoy reading a famous book called The Mythical Man month.

https://www.amazon.com/dp/0201835959/?tag=pfamazon01-20

His point is that adding people to a project often makes the project take longer! Because you have to now supervise and coordinate more people.

Another famous example is attempting to use nine women to make a baby in one month :-)

 

[ViolentCorpse]

You might enjoy reading a famous book called The Mythical Man month.

https://www.amazon.com/dp/0201835959/?tag=pfamazon01-20

His point is that adding people to a project often makes the project take longer! Because you have to now supervise and coordinate more people.

Excellent point! I did mention that, but the irony is that, he is supposing a very hypothetical situation and at the same time dismissing mathematics because it isn't matching reality.

Another famous example is attempting to use nine women to make a baby in one month :-)

Hahaha! Good one! :D

 

[ArcanaNoir]

The problem isn't that math doesn't represent reality, its that the equation considered was insufficient. You should be looking for the limit of how fast the building can be built as the number of workers approaches infinity. Since there are things like the amount of time the cement will take to dry, the number of workers that can be applied to riveting one steel beam, etc, the actually equation representing how fast you can build a building is far more complicated than simply dividing the average time by the number of workers. Does that make sense?

In summary, its not the math that doesn't represent reality, its the mathematical model you are using.

 

[cmb]

He asserted that Math can give you results far-off from what's real. He illustrated his claim like this: Suppose you want to build an average-sized building. You employ some workers for the job and and then estimate mathematically the time the building should take to complete. You get a certain estimate but it occurs to you that if you employ more workers, it should take less time. Suppose then that if you add just enough workers, your calculation would give you a time estimate to complete the building of 5 minutes. But that is impossible in reality. No matter how many workers you add, you can not build and average-sized structure that fast!

This is not a question about the maths, but about the mathematical model.

In fact, this isn't to do with 'maths' at all, but to do with 'mathematical modelling'. Here, your model is that the time, T, it takes to complete a building with N workers is k = T.N (where k is a constant related to the building details).

What you have concluded with your friend's thought experiment is that the model k=T.N is wrong. That is all.

I suppose it is best to halt the discussion there, but there are some other lines of discussion available. For example, you might be wondering is maths good for anything, then? Well, clearly the model k=T.N is going to be moderately accurate for some range of values but not for all. So what happens (has happened) in the real world is that people build their first building and derive a value for k. Then they build a second just the same, and find it is not quite k. This now begins to look at the statistical accuracy of that model. Then they build a building twice as big and conclude a second number j such that for a building size j then j.k=T.N , &c..

The thing is, a model is only a model and it is a whole subject on its own (within physics rather than maths) to determine how, when, where and what is the appropriate derivation and use of a model. Is it useful to do this? Well, put it this way, you might might not be sitting warm infront of your computer reading this, rather still be walking barefoot in the dark to your weaving job, if it wasn't. Possibly, you might not even be that lucky.

 

[ViolentCorpse]

The problem isn't that math doesn't represent reality, its that the equation considered was insufficient. You should be looking for the limit of how fast the building can be built as the number of workers approaches infinity. Since there are things like the amount of time the cement will take to dry, the number of workers that can be applied to riveting one steel beam, etc, the actually equation representing how fast you can build a building is far more complicated than simply dividing the average time by the number of workers. Does that make sense?

In summary, its not the math that doesn't represent reality, its the mathematical model you are using.

Absolutely! He narrowed it down way too much and, I've thought about that cement thing too! The problem is that he made a model that was far-off from reality and blamed math for being the same.

The bigger reason why I created this thread is because it made me think about the nature of theories in physics. Relativity for instance. It has always been considered a good scientific theory from the start. Of course it was mathematically perfectly consistent and the model was probably a good one. But does that alone justify it being considered a good scientific theory at the time when no tests of it were performed?

 

[cmb]

But does that alone justify it being considered a good scientific theory at the time when no tests of it were performed?

It made a prediction that was contrary to the existing science, yet could be tested. Generally, that makes it an interesting and 'good' theoretical proposal.

 

[Number Nine]

Mathematics is a language; its purpose is to express very complex concepts in a clear and unambiguous way. You are essentially asking "Does language always correspond to reality?". The answer is, of course, no; it only reflects reality if what you are saying reflects reality.

 

[ViolentCorpse]

Thank you for all the replies, people! I really appreciate it!

I have got another question raised in my mind; We often run into problems that have more than one possible solution. Sometimes all the solutions are equally possible, other times we run into a solution that can not be true for the purpose we're trying to solve it for. For instance, we find that 'x' has two possible values: 3 and -3. Now, if our problem was to find the length of something, then obviously '-3' can not be the right solution, since, in our world at least, length can not be a negative number. But perhaps for some dude in a different universe doing the same problem, it might be that the '3' solution for x is the wrong one.

What I'm trying to say is that, Math only gave you the solutions, it can not tell you whether they're both good for our purpose or just one of them is. So does that mean that our solutions need interpretation? Or am I missing something here (as I often do)?

 

[bhobba]

No. Check out the Lorentz-Dirac equation - it famous for having solutions that are just plain silly:
http://arxiv.org/PS_cache/gr-qc/pdf/9912/9912045v1.pdf

Thanks
Bill

 

[cmb]

So does that mean that our solutions need interpretation?

Of course! Mathematical models are tools that we have devised for us to interpret the world, therefore it is only as good as our interpretation and our ability to devise, and need monitoring to ensure their performance is as we need.

Hammers and nails were invented before screws. If folks hadn't monitored how that was working and ask themselves if other means were appropriate, and devised a new way of fixing things together, we'd not have screws and screwdrivers! That doesn't mean hammers and nails are no longer useful, just that they have their place. And someone needs to decide whether to use a hammer, a screwdriver, or to devise a new tool when they start a job. This is no different to mathematical modelling.

 

[Deveno]

no. math rarely corresponds to reality, usually the physical system is SO complex, that its beyond our abilities to even construct a faithful model.

but, when making a decision, does one ever consider every single relevant fact, from the present or any point in the past, that might be a factor? we'd be in a constant paralysis of indecision. i can't decide whether or not to wear my raincoat, until i have the exact location and velocity of every single water molecule in the atmosphere?

no, we settle for approximations that give us useful decison-making information. we "forget stuff". sometimes, if we lose track of the assumptions we're making, we run into the "garbage in, garbage out" syndrome. our models need to be tested aginst our empirical observations for consistency.

a simple example: Betty's house is 4 blocks east, and 3 blocks north of her school. If she can walk 1 block per minute, what is the soonest she can arrive home, after school let's out?

well, we start thinking: the shortest distance between two points is a straight line. put her school at (0,0) and her home at (4,3). calculating the distance between her school and her house, we invoke the pythagorean theorem:

D² = 3² + 4² = 25.

since the solutions to this are D = -5, and D = 5, and since she can walk 1 block a minute, we conclude that the soonest she can arrive at home, is 5 minutes before school let's out. wait, what?

there's nothing "mathematically" wrong about taking D = -5, but physically, it corresponds to some house 4 blocks west, and 3 blocks south, of the school (presumably Betty doesn't live there, too). in other words, the process is something like this:

A (physical problem) ---> model ---> B (math problem).

B ---> (math stuff) ---> C (math solution)

C ---> application ---> D (physical solution)

the B-->C part is usually iron-clad, but the correspondence A-->B usually ISN'T 1-1. so we have to be careful, when we go C-->D that we "interpret" the results appropriately (there's often multiple interpretations, we need "extra real-world information" to help decide WHICH interpretation we go with).

in the case of our wandering school-girl Betty, what we mean to say is that we are only interested in "positive" distances (towards her home, not away from it). the math gives us two choices, we have to consult the way we built our model, to tell us which one we want, the math doesn't say.

this happens all the time when we solve differential equations. often we have some initial condition that has to be satisfied, as well, that tells us "which" solution that the math spit out "applies".

*****

all that said, we have no reason to believe that math is "real". the best we can actually say, is that math is an economical way of describing what we observe, and is subject to some of the same linguistic difficulties of any language (at some point, some terms are "undefined", we don't know (or can't say) what they mean, but we know how to use them). you may choose to take comfort in the notion that objective reality exists, and that mathematics describes it, but such a notion is philosophy, it is neither mathematics nor science.

 

[ViolentCorpse]

no. math rarely corresponds to reality, usually the physical system is SO complex, that its beyond our abilities to even construct a faithful model.

but, when making a decision, does one ever consider every single relevant fact, from the present or any point in the past, that might be a factor? we'd be in a constant paralysis of indecision. i can't decide whether or not to wear my raincoat, until i have the exact location and velocity of every single water molecule in the atmosphere?

no, we settle for approximations that give us useful decison-making information. we "forget stuff". sometimes, if we lose track of the assumptions we're making, we run into the "garbage in, garbage out" syndrome. our models need to be tested aginst our empirical observations for consistency.

a simple example: Betty's house is 4 blocks east, and 3 blocks north of her school. If she can walk 1 block per minute, what is the soonest she can arrive home, after school let's out?

well, we start thinking: the shortest distance between two points is a straight line. put her school at (0,0) and her home at (4,3). calculating the distance between her school and her house, we invoke the pythagorean theorem:

D² = 3² + 4² = 25.

since the solutions to this are D = -5, and D = 5, and since she can walk 1 block a minute, we conclude that the soonest she can arrive at home, is 5 minutes before school let's out. wait, what?

there's nothing "mathematically" wrong about taking D = -5, but physically, it corresponds to some house 4 blocks west, and 3 blocks south, of the school (presumably Betty doesn't live there, too). in other words, the process is something like this:

A (physical problem) ---> model ---> B (math problem).

B ---> (math stuff) ---> C (math solution)

C ---> application ---> D (physical solution)

the B-->C part is usually iron-clad, but the correspondence A-->B usually ISN'T 1-1. so we have to be careful, when we go C-->D that we "interpret" the results appropriately (there's often multiple interpretations, we need "extra real-world information" to help decide WHICH interpretation we go with).

in the case of our wandering school-girl Betty, what we mean to say is that we are only interested in "positive" distances (towards her home, not away from it). the math gives us two choices, we have to consult the way we built our model, to tell us which one we want, the math doesn't say.

this happens all the time when we solve differential equations. often we have some initial condition that has to be satisfied, as well, that tells us "which" solution that the math spit out "applies".

*****

all that said, we have no reason to believe that math is "real". the best we can actually say, is that math is an economical way of describing what we observe, and is subject to some of the same linguistic difficulties of any language (at some point, some terms are "undefined", we don't know (or can't say) what they mean, but we know how to use them). you may choose to take comfort in the notion that objective reality exists, and that mathematics describes it, but such a notion is philosophy, it is neither mathematics nor science.

Wow! That was very comprehensive!

That is exactly the role I've believed Math to be attributed to for most of my life. But recently, I started reading a lot into things and it got really fuzzy. But this thread has reassured me.

Thanks, Deveno! That was an excellent post!

Edit: Oh and, regarding the example you gave me, one can make the objection that Betty probably won't be able to walk in a perfectly straight line to her home. She might find things in her way that she'll have to go around. And in that case, the solutions we found won't hold true for our physical conditions. But that is obviously not the Math's fault, but the assumption that we made in the first place that she can walk in a perfectly straight line to her home. So in fact, it was the model which was averted from our physical situation!

Am I understanding this right?

 

[Deveno]

yes, and there are other considerations, too: the surface of the Earth is curved, and her path isn't perfectly smooth, she takes quantized "steps", and she is unlikely to travel at a uniform rate of speed.

so, for example, in calculating something like a trajectory of a projectile though the air, you have irregularities introduced by the coefficient of drag, air density, wind velocity, nearby magnetic fields, etc. in the model, you make the assumptions that these effects are small compared to the large effect of gravity, or (in the case of a self-guided missle, let's say), you introduce some kind of error-correction factors. even the relatively simple problem of calculating a near "ideal" object's movement under carefully controlled conditions can get pretty intricate fairly quickly (the standard example is a simple harmonic oscillator, i believe, with attendent "complications" of driving or dampening): witness the difficulty inherent in generating a realistic-sounding "guitar sound" from a synthesizer.

in a nutshell, the modelling standard for correctness (or appropriateness, a better term, in my opinion) is its agreement with observation. if we arranged a large-scale "mock-up" of the city blocks containing Betty's school and house (but with no buildings) and perhaps appropriately programmed robotic vehicles that always moved at a speed of 1 block/minute, and then we found that some robot consistently reached the "house" from the "school" in less than 4 minutes, we would conclude that our model was fundamentally flawed, and try another approach, rather than: "oh, math is useless!"

 

[ViolentCorpse]

Great! I think I understand.

Thanks to all the people who contributed here. All the responses were very helpful. I really appreciate it. :)