Why is mathematics so effective when describing nature?

[Erik Mesoy]

1 apple and 1 apple always gives 2 apples, independendly of symbols and languages - thinking about it...- we should send them some apple-math books to establish contact

Boring regurgitated answer - define an apple. If I slice an apple apart with a knife, do I have two of ½ an apple? In that case, since ½+½=1, how come I can put the halves together and they don't form an apple? Does the apply include the stalk? Can I cut 1/1000000th out of each of 1000000 apples and have a new apple? Can I cut a molecule out of each apple? How about if I synthesize an apple from scratch?



In short, it's not independent that way at all - rather, mathematics is independent of apples because mathematics describes absolutes and most other people work in approximations.

 

[warpus]

In a universe where an observer is often needed to "collapse the wave function", can it be said that logic exists on its own?

Shrodinger's Cat implies that if that were the case, you would need an original observer (I'll give you one try to figure out who that observer would need to be...)
As I have no need for that particular hypothesis, the other, less appealing notion (that a theorem exists only after it is formulated) merits some consideration. And that's the crux of the whole "debate."

That's if you take the Copenhagen interpretation of quantum physics. You could take the less-favoured interpretation of the double-slit experiment and claim that the theorem exists in all parallel Universes, even before it is observed/formulated.

Humans use mathmatics to describe or characterize physical phenomena such as flight. We use all kinds of equations to do so. Birds have been flying for millions of years. Are those equations anything more than just our view/perspective of what birds do? Are they anything more than just flight seen through the filter of human reason?

Exactly. The mathematics we use to explain things such as flight is just that - an explanation of an event that exists outside/independent of this explanation. It's the same for theorems - the logical ideas behind the theorem were 'true' before a human discovered and proved the relationships.

A staple of science fiction is that aliens would use math as a common ground to communicate with us. Anyone reading this thread now realises that's a huge assumption.

IMO it's such an unreasonable assumption. Before aliens are able to build spaceships they have to explain how many things in the Universe work... somehow. In order to do that you need some sort of a system that's based entirely on logic - that attempts to approximate and predict the behaviour of various systems in the Universe. It surely wouldn't be exactly like our Mathematics, but unless the laws of physics were different in their part of the Universe, there would be many similarities.

Boring regurgitated answer - define an apple. If I slice an apple apart with a knife, do I have two of ½ an apple? In that case, since ½+½=1, how come I can put the halves together and they don't form an apple? Does the apply include the stalk? Can I cut 1/1000000th out of each of 1000000 apples and have a new apple? Can I cut a molecule out of each apple? How about if I synthesize an apple from scratch?

In short, it's not independent that way at all - rather, mathematics is independent of apples because mathematics describes absolutes and most other people work in approximations.

When we use mathematics to model events that occur in the Universe, we realize that it is simply going to be an estimation of what occurs in real life. It won't be an exact representation of what is going on - but it's going to be a useful representation nevertheless.

So if you're trying to count whole apples, you first define what an apple is. Once you've done that, you can go ahead and add apples to apples all you want. If you cut an apple in half and wonder how that affects your calculations - you're going to have to define some sort of an apple-fraction system so that your calculations make sense.

 

[Che Guava]

What a great little thread!!! I have been thinking about this stuff a lot lately (it tends to happen when you work fanatically on statistics...).

My $0.02: I side with Einstien. Mathematics are are as objective and abosulte as you need them to be, the only restrictions are what we can observe (i.e. the two halves of the apple) and the language in which we communicate those concepts.

Why do maths still seem objective to us? Mostly because we are discovering that the complexity of phenomena that we see around us is near infinitely complex, to the point where we can only describe it as random.

Take the weather: from our perspective (that being our size, the length at which we live/perceive time, the scale at which we are affected by it), we can predict weather patterns based on large trends: pressure changes, wind currents, etc. But if we want to know if it will rain tomorrow, we can't say for sure, because that depends on too many variables at too minute a level to be able to predict for sure. With enough data and a sophisticated model, you could probably predict the weather for tomorrow with near abosolute certainty, but it would probably involve tracking the movement of molecules across many square km, along with lots of other stuff that would just take too much time and effort.

Exactly. The mathematics we use to explain things such as flight is just that - an explanation of an event that exists outside/independent of this explanation. It's the same for theorems - the logical ideas behind the theorem were 'true' before a human discovered and proved the relationships.

I guess we're the only species here that tries to figure out 'why'

IMO it's such an unreasonable assumption. Before aliens are able to build spaceships they have to explain how many things in the Universe work... somehow. In order to do that you need some sort of a system that's based entirely on logic - that attempts to approximate and predict the behaviour of various systems in the Universe. It surely wouldn't be exactly like our Mathematics, but unless the laws of physics were different in their part of the Universe, there would be many similarities.

My thoughts exactly. It has been proposed that the biggest barrier between us and alien life might be temporal and scale differences. What if these aliens are the size of planets? What if they live for only 5 minutes? How do you explain to someone your biology, for eg, when a peice of your DNA is as small to them as an electron is to us? Mathematical rules apply no matter what the scale, and I'm sure no mater what we meet 'out there' , math will be our first contact...

When we use mathematics to model events that occur in the Universe, we realize that it is simply going to be an estimation of what occurs in real life. It won't be an exact representation of what is going on - but it's going to be a useful representation nevertheless.

So if you're trying to count whole apples, you first define what an apple is. Once you've done that, you can go ahead and add apples to apples all you want. If you cut an apple in half and wonder how that affects your calculations - you're going to have to define some sort of an apple-fraction system so that your calculations make sense.

Precisely. What I'm finding out is that a big part of finding major trends in systems is changing your own perspective. For my thesis right now, I'm using some really interesting multivariate stats techniques that basically plot out points along multiple axis' (more than 3D...) and then moving around your own perspective to find the best 'vantage point' to explain what you are seeing. Its kinda similar to, say, viewing the milky way from as many different plants as posible to find the one that will give you the best picture...

 

[newfangle]

I have trouble even believing that math is a good representation of reality.

Solutions to PDEs (for instance) remain decent approximants of reality, but not dictators of it.

But then, isn't that exactly what applied mathematics is?

But then I see Hilbert modules being used to explain quantum computing and I get confused again.

Maybe I'll just settle with "math is great and anyone that doesn't know it is a big idiotface." (of course I'd still be a giant idiotface).

 

[Che Guava]

I have trouble even believing that math is a good representation of reality.

But math IS reality . THat's the wonderful thing about it: it's the most basic tenant (that we know of) of the laws of the universe. No matter how you might define them, 2 + 2 always equals 4 regardless of size, shape, or anything else.

Mathematical models, OTOH, aren't reality. They are tools for obseving a system by reducing the scale and complexity to a level that we can understand. It brings us closer to a vantage point that makes sense to us. Take the solar system for example: we can leanr from it by directly observing everything that happens, or we find the major changes and cycles it goes through, then change the spatial and temporal scale to match ourselves,

FOr eg,instead of being an ant trying to describe a skyscraper by sitting on the first floor for a day, we reduce the size of the building to something we can easily view, then increase the rate at which things occur to synch up our own 'processing speed' with the rate of change of the building. It makes for a somewhat crude represenation because the model is based only on what the ant could see on the first floor for a day, but in that context, the model does provide us enough to understand the underlying processes of the system.