Mathematics and the real world

[rjb]

I've never really understood math. Not beyond the point of making some simple calculations. All mathematicians argue that "math is the true language of nature", but I really can't see the connection. I can't imagine what they're talking about and how do they figure out stuff in math and be so sure that the real world response will be exactly as predicted?

Can someone please explain, in layman terms, how to translate a real life situation into a mathematical formula? I can only think of simple things, like:

I go to the supermarket having $50. I need to buy some laundry detergent ($20), bread($2) and butter($5). Do I have enough left to go to the movies afterwards, given that a ticket costs $15?

This would be (I assume): x = 50 - (d + b1 + b2) <=> x = 50 - (20 + 2 + 5) <=> x = 23. So yes, I can go to the movies.

But I'm more curious about how more complex situations translate into math. Like, for example, I recently watched a documentary about space, and at some point they said that through math, they figured out that a black hole, when swallowing an object, stores its digital information on top of that black hole, even though the object itself is destroyed. I'm extremely curious how do they know that it's not an aberration? We don't have a black hole handy, so how can you figure THAT out? Anyone could easily come up with stuff like that, without knowing any math at all. So, how do they reach such conclusions?

Another weird one is Einstein's equation, E = mc2. I can't figure out why he felt he needed to square the cosmological constant, as the speed of light is [measurably] the same all the time and nothing can exceed it. Why not just E = mc?

I'm not necessarily asking for explanations to these questions, but rather for a general pointer as to where to start understanding this relation between math and real world situations. How would a meeting between 2 people be represented into a math formula? How would you find out, through math, if you really are your parents child? How would these things be formulated into a mathematical model?

Please help me figure out this stuff. I'm sure I'm not the only frustrated monkey

Thanks.

 

[Ovidroid]

Asking for an explanation of the Einstein field equations in a forum like this, or really any forum for that matter, is probably not the best way to get an understanding of black holes or the consequences of general relativity. If you want a better understanding, without all the advanced mathematics that you would need years to understand, you are better off going finding a source that will put it in layman's terms.

I am no mathemetician, I am just an engineer, so I can't explain those equations for you, and I don't know how much I can really help you with math, but I will say that if you want to get a pretty decent grip on the physics of the world around you, your best bet is to learn calculus.

If you understand algebra very well, if you know trigonometry, and if you can learn some new symbols, calculus really isn't conceptually all that difficult. Fundamentally, calculus is the mathematical way to describe change. For example, if you are driving down the road, you have a position, a velocity (which implies direction and speed), and an acceleration. Much of calculus involves differentiating and integrating functions that describe the relationship(s) among different variables. In the driving example, your velocity is the derivative (the result of differentiation) of your position with respect to time, meaning that if you describe your position as a function of time, say, like

Position(t) = 70*t

your velocity will be (as you remember) distance over time, except that when you take the derivative it is, ideally, valid for any time t you happen to be at. So, rather than just being able to calculate your velocity at one point, if you know how your position varies with time, you can figure out your velocity at ANY time (meaning at any position in your journey)..Suppose we are using units of miles per hour, and time t is in hours. If you differentiate your position as a function of time, you end up with this:

d(Position)/dt = Velocity(t) = 70 mph in this example the 't' just drops out

d(Position)/dt just means you are taking the derivative of your position with respect to an infinitessimally small interval of time dt. It's essentially calculating the slope of your position function (remember rise/run?). If your position function were:

Position(t) = 5*t^2 + 13*t - 2

your velocity at any time t would be calculated as

d(Position)/dt = 10*t + 13

Now, just like velocity describes your change in position with respect to time, acceleration describes your change in velocity with respect to time:

Acceleration(t) = d(Velocity)/dt = d^2(Position)/dt^2

This means acceleration is the first derivative of velocity and the second derivative of position.

Using the first example, since you are cruising along at a constant speed, your acceleration would be zero. The derivative of a constant is always zero.

Using the second example, your acceleration would be:

d(Velocity)/dt = 10 (because the 13 is a constant and its derivative is zero, and the derivative of a variable having an exponent and a coefficient is coefficient*oldexponent*variable^(old exponent -1))

I'm not trying to make it confusing here. There are rules for how to differentiate different types of functions. My point is simply that if you want to go beyond just calculating your grocery list and how it will financially affect you, you should look into calculus : )

I can't imagine this has been much help. It's a VERY elementary description of calculus, and I left out lots, including integrals which are essentially the opposite of derivatives, simply speaking. But, math is the language of physics, and I was just trying to show that with a simple example.

As for E=Mc^2, it comes from physical assumptions combined with relativistic principles (physical assumptions such as the conservation of energy). Just to point out why you wouldn't say E=Mc, it is a unit cancellation issue (other than the fact that when you derive E=Mc^2, you don't just decide to make the speed of light squared it comes out of the math). If it were just E=Mc the units on the right side of the equation wouldn't match the units on the left side of the equation, and the equation would just not make any sense at all. It's like the driving example, if you have the following

Position(t) = 5(miles)*t

your velocity is (incorrectly) written as 5 miles (you know it would be 5 miles per hour), so, if you are measuring t in hours, your incorrect position description would now be say at 5 hours, your position would be 25 mile*seconds. Doesn't make a whole lot of sense, right? Same thing with any equation involving physical units.

I know these examples aren't very exciting, but calculus is used for everything from the mundane to the spectacular. Without it, math could only describe a static universe. With it, things can CHANGE!

 

[benzyme]

1 = 2

 

[Crazyhorse]

Real world maths huh?

well, If you add 67,740 to 12,345, it looks like it says BOOBS on a calculator screen.

That's all I got.

 

[John Smith]

It's hard to give a big picture, I guess examples from more applied branches may kinda apply. Like after doing some fairly higher level mathematics you can end up proving some results on the inner working of the world(like law of large numbers). Keep tossing a fair coin, you'll end up with equal amount of heads and tails in the long run. That is obviously a very simplistic example... nevertheless.

Some areas, like pure math, don't seem to be applicable(borel set to real life? lol)... until it is used in other fields.

 

[Guyomech]

The bottom line is that math is an internally consistent system, with rules that apply consistently across tge whole thing. This allows math to be used as a means of comparing measurements of real world phenomena and ultimately drawing conclusions about them. The reason our maths have been so successful is that the results derived through calculation tend to match real world observation. You use math to calculate the amount and duration of thrust needed to get a payload of a given weight into a given orbit; do your math right, engineer the rocket to deliver the exact amount of thrust called for by those calculations, and you will successfully put it in that orbit. Math has been used to successfully predict many complex things such as gravitational lensing (another Einstein concept) which then turn out to match what happens in real life. Most rational people agree that this is proof that mathematics can be used to make valid models of real life.

But you can't use simple equations to describe complex human interactions and things like that. It's mainly useful for exploring the mechanics of the universe.

Another interesting point: mathemetician Kurt Godel created a proof that showed that our math system is ultimately incomplete. Basically, he used provable statements to form the overall statement; "this statement is unprovable". This internal contradiction gave a lot of math people fits but nobody was able to make it go away. I like to think of this as an example of how we humans, from a vantage point within the cosmos, can never truly have a complete understanding of our big picture, at least not one that can be proven.

 

[SmoovPnCali]

Real world maths huh?

well, If you add 67,740 to 12,345, it looks like it says BOOBS on a calculator screen.

That's all I got.

Except for this response....I'm getting a headache...

 

[Mr.Peabody]

Check out the series of vids on how plants and math are related:

her videos are very good, entertaining, and put in understandable terms.

I never knew how much I loved math until I started taking it for my engineering degree. I especially love the physics of Newton, such as falling bodies in gravity, and projectile motion. Ovidroid did a very good job of explaining some of these types of things.

You can buy calculus and physics books for very cheap, and can actually teach yourself a lot. I learned this by having some pretty bad professors. It may turn out to be something you had no idea you would really find fascinating!

 

[corpus callosum]

Another interesting point: mathemetician Kurt Godel created a proof that showed that our math system is ultimately incomplete. Basically, he used provable statements to form the overall statement; "this statement is unprovable". This internal contradiction gave a lot of math people fits but nobody was able to make it go away. I like to think of this as an example of how we humans, from a vantage point within the cosmos, can never truly have a complete understanding of our big picture, at least not one that can be proven.

Wise words from Guyomech, IMO.

For those not aware of Godel, his Incompleteness Theorem is interesting stuff and worth being aware of (again, IMO).Heres an explanation of it, in laymans terms, followed by a long and at times heated discussion about it and its implications for other issues that seem to be a regular topic of discussion here at the Nexus:

godels-incompleteness-theorem

 

[corpus callosum]

Another interesting point: mathemetician Kurt Godel created a proof that showed that our math system is ultimately incomplete. Basically, he used provable statements to form the overall statement; "this statement is unprovable". This internal contradiction gave a lot of math people fits but nobody was able to make it go away. I like to think of this as an example of how we humans, from a vantage point within the cosmos, can never truly have a complete understanding of our big picture, at least not one that can be proven.

Wise words from Guyomech, IMO.

For those not aware of Godel, his Incompleteness Theorem is interesting stuff and worth being aware of (again, IMO).Heres an explanation of it, in laymans terms, followed by a long and at times heated discussion about it and its implications for other issues that seem to be a regular topic of discussion here at the Nexus:

godels-incompleteness-theorem

http://www.perrymarshall.com/articles/religion/godels-incompleteness-theorem/

 

[rjb]

1 = 2

Very much liked that, benzyme. So would you say math is inherently flawed? It's not like it's an impossibility, just more of a wonderment(ing).

Real world maths huh?

well, If you add 67,740 to 12,345, it looks like it says BOOBS on a calculator screen.

That's all I got.

Yeah :lol:

Ovidroid, thanks a lot for the hands-on explanation. Actually, that made some logical sense, but I do have to look into calculus further so I can totally understand the process. I think where math loses ground is in this logical => mathematical => logical conversion. No one really explains the mindset for those kinds of things, but I think this will start to emerge with practice.

Lots of info to assimilate, I particularly liked the Fibonacci example, which I've definitely seen before and seems like the easiest function to assimilate, due to its closeness to something easily observable. We need more of those videos. And now I have some more knowledge for drawing, too :lol:

Thanks for the Godel theorem, looks like an interesting POV. Will definitely look up all the info on it.

Well, looks like things are starting to make some sense. A little, but it's something compared to being completely oblivious. Excitement for math starts to reappear in my picture. Yay!

 

[EthanS]

Hi,

Please note that the following is my point of view only and in no way do I claim this to be *the* way or what *maths is really about*. It is merely what I came across in my own mind to explain to myself over the years similar questions as the OP. I might also be completely wrong but the information below, even partial or incorrect might be useful.

What I understand is that maths and numbers are not necessarily linked and this causes the confusion as the grocery mathematics and financial mathematics are not the same as the mathematics used for physics or other domains. If I am not mistaken the *model* of the decimal mark (point or comma depending on countries) is relatively a recent invention - can you imagine doing financial calculations without having decimals and using only fractions and the sort?

As such, *mathematics* in the strictest sense don't actually exist and the word *mathematics* is applied to a broad spectrum of other (real) domains which look similar, but aren't the same actually.

Most of the times what from far away looks like "maths" is in fact just *logics*, meaning the use of symbols (mainly letters, latin and greek) in a *logical* way as to preserve the truth of the starting ingredients. As long as the connections and transformations are logical, thus the truth preserved, any outcomes and conclusions, no matter how obvious or outrageous will also be logical and true. The necessity to do this with logics (maths) is because the things investigated have many variables and components, and putting things down on paper with the appropriate symbols and logic rules, allows the exploration of the topic without being concerned about human error where there would be slips or little lies and distortions would make their way into the topic and alter the end result.

The purpose of the model of maths/logics is to be deterministic. Meaning doing the same thing two times, with the exact same starting variables and rules will result into the exact same result. This is very very close to nature but not a perfect match, as when things in nature get very big or very small the deterministic aspect doesn't hold and we end up with the concept of random, or the necessity of parallel universes to explain quantum mechanics etc.

I find it useful to think of the use of logics (and maths) as being *practically* very useful and precise, but at the same time don't get surprised if one crucial day it just won't work for a particular topic, as these logics and maths are OUR (humans) logics and maths, a model, and are not god's word.

Now this being said I must also point out that *maths* are not some special domain that does magic but everything is in fact a string of domains that link up together. Before logics, there is philosophy that starts the assumptions; philosophy is also logical. There is also geometry (although numbers are used in geometry they are not necessarily linked). Art as well has logics and links with geometry etc.

May I recommend to look into geometry - more precisely the sacred geometry domain - Flower of Life, as this will outline the point how *geometries* exist as a real aspect of reality.

Also may I point towards the work of Marko Rodin which provides a very nice point of view about the nature of numbers and why calculations, multiplications, etc. work the way they work.

Kind Regards,

Ethan

 

[Kensho]

All mathematicians argue that "math is the true language of nature"

I don't agree; very few mathematicians I've learnt from have made such claims. Mathematics is a game, where we suppose that certain axioms are true and see what follows logically from those suppositions. Then we define new objects and constructs. And we define them by what they do, not by what they are. Why? Because doing so is convenient. And entertaining. But as a tentative mathematician I can't say that I see any "truth" in the subject. Not more that what I see in a game of cards, anyway.

People into physics, however; now, they're often quite a different story...

 

[John Smith]

All mathematicians argue that "math is the true language of nature"

I don't agree; very few mathematicians I've learnt from have made such claims. Mathematics is a game, where we suppose that certain axioms are true and see what follows logically from those suppositions. Then we define new objects and constructs. And we define them by what they do, not by what they are. Why? Because doing so is convenient. And entertaining. But as a tentative mathematician I can't say that I see any "truth" in the subject. Not more that what I see in a game of cards, anyway.

People into physics, however; now, they're often quite a different story...

Nicely put.

Math is a system that is as good as humans make it, basically. Back maybe a hundred years ago or more, some guy went on to prove how many axioms/definitions are not rigorous/precise enough by contradicting large amounts of theorems and coming up with all sorts of inconsistent results - from there on people basically went on to reinvent/redefine existing definitions(rigorous definition of a limit(which might seem excessive to some) is one example). So I guess one could argue that we just operate by human-invented system that might not necessarily be true. Anyone who hasn't studied rigorous intro to basic math I'd highly recommend viewing this PDF, it just defines basic properties of numbers - you can see that pretty much everything lies under assumptions. Such as one unit + 0 unit = still equal 1 unit. Statement is either true, or false... etc

http://math.nyu.edu/~tsang/classes/precalc1/session1.pdf

Its funny from my point of view but from these basics we end deducing that upon absorption into black hole only leaves information and other crazy stuff rjb mentions.

 

[JacksonMetaller]

I don't think you're going to find the answers you're looking for on here. Mathematics are very complicated and to truly understand how they apply to the universe you're going to have to go very deeply into it.

But to answer your general question, mathematics is just a translation of the logic of the universe so that we can understand it. As any translation, the full depth of the situation doesn't always get carried over. But if you can imagine two fundamental particles (more fundamental than quarks or anything like that. something more along the lines of string theory where everything is made up of the same one thing vibrating at different frequencies/having different charges/however you want to imagine it), then imagine the interaction of those two particles with each other. It would be something very simple, say like "positive and negative attract" or something of the sort (no one really knows this as this fundamental component has not yet been found). But you could see how two very small particles could react with a very simple logic. The equation you would write for it would probably be no longer than an inch or so. But say you have another two particles that interact with each other and then interact with the original two particles. You now have a slightly more complex relationship, but it's all governed by the same initial logic. This continues until things are so far evolved they no longer resemble each other. However, you don't want to start from ground zero every time you want to figure out how to send satellites into orbit (nor do we know where ground zero is yet) so you derive new equations from the original logic which help you understand the more immediate situation. I'm not great with math so I will give you a chemistry example. Say you want to know why water sticks to your skin. Someone tells you it's because it's polar. But voodoo magic doesn't happen in the universe. There's a REASON why it's polar. So then you have to understand what makes things polar. You then reason that it has to do with the electronegativity of oxygen and hydrogen. Sufficient answer? No. What causes differences in electronegativity? Electron shell configurations. What makes those act the way they do? And so on until you reach the fundamental principle that governs everything. The thing that has no explanation but itself and is responsible for everything you see. But if you just want to understand polarity do you need to go back to the fundamental principle every time? No, you create laws and understand trends to be more efficient.

So though it seems like equations are just taken out of the blue they're not. Everything in the universe is governed by a very complex order derived (I believe) from one simple line of logic. Because things are always following this same logic they're very predictable and you can derive equations to explain their behavior.

Keep in mind this is an example based on my view of the universe. No one has proven anything beyond quarks and at the moment we have many fundamental particles, not one. Though, I think many physicists agree that in order for things to be balanced and logical, one fundamental component seems like the most rational answer. Otherwise everything would just be random numbers. That's why they're unsatisfied with the gap between quantum mechanics and general relativity. Because we're pretty certain that all actions are connected by a more fundamental principle. Once you understand this, you can understand everything. But until then we just abstract equations based on observation.

 

[EmptyHand]

Mathematics is a game, where we suppose that certain axioms are true and see what follows logically from those suppositions. Then we define new objects and constructs. And we define them by what they do, not by what they are. Why? Because doing so is convenient. And entertaining. But as a tentative mathematician I can't say that I see any "truth" in the subject. Not more that what I see in a game of cards, anyway.

This is a philosophical view on mathematics called "Formalism" which is approximately 100 years old and almost no working mathematician takes seriously anymore as an accurate description of the practice of mathematics or the nature of mathematical objects. (Incidentally, I am one of those practicing mathematicians who thinks this is a grossly distorted view.)

The mathematical objects that get studied, in contrast to toy constructs that simply get thrown away, are those objects that get into our face and shout, "I am VITAL. I am important to REALITY." The importance can originate in physical applications or elsewhere, for example mathematics itself.

If you think the primary function of mathematics is to serve the physical sciences than you probably have not progressed beyond calculus or linear algebra yet. Calculus is just the very first step into non-trivial mathematics. Drink deep.

Godel's incompleteness theorems, like quantum mechanics, is vastly overused in philosophical discussions for their supposed implications about the fundamental nature of reality. The incompleteness theorems simply describe limits to one tool in the mathematician's arsenal, axiomatic systems, but not limits on mathematical objects themselves. As an analogy, particle accelerators are useful to physicists for some investigations but inadequate for other pursuits in physics.

Finally, axiomatic description of a mathematical object represents a mature phase of research and only occurs after many years of intuitive understanding. For example, Poincare understood what a topological space is long before the axiomatic description of topological spaces was specified.

Thus, axioms DESCRIBE mathematical objects but do not CREATE them. This is a form of mathematical realism (grounded in phenomenology) but is not Platonism.

eH