Atheism and the Afterlife

smacl

starshine1234 wrote: »

After a lot of thought I've decided that maths is more than a description.

Good for you. So perhaps name something that can be described in mathematical terms that can't be similarly described long-hand in English? Mathematical expression is the most practical way of describing certain things, and an impractical way of describing others. Humankind has created various forms of mathematical expression much as it has created many other languages and forms of expression, but it is a mistake to confuse the description with the thing described. For example, Newtonian physics describes our universe at a macro level but fails to describe it at a Quantum level. FWIW, my understanding of maths would follow a formalist school of thought.

And on a lighter note

starshine1234

It is the mathematical truths themselves that have an independent existence, not the description of them. The mathematical truths transcend their own descriptions, and the physical universe.

I agree that the maths notation we use could change. That is no different from saying we could use a different human language to describe the same thing, like French, English or Mandarin for example.


The area of a circle is pi.r.r = (pi r squared)
That is true even if we have no language to describe it with, or no universe for the description to exist in.
The truth itself has an independent existence, and transcends all other things.


I don't believe it's correct to say that water is wet if there is no universe and no humans and no human language.


The truth that water is wet depends on our universe, because the english language depends on our universe, and water itself depends on our physical laws.
The truth that the area of a circle is 'pi r squared' does not depend on our universe, and that truth transcends all universes.

That seems to be a difference but maybe it's just semantics.

smacl

starshine1234 wrote: »

The area of a circle is pi.r.r = (pi r squared)
That is true even if we have no language to describe it with, or no universe for the description to exist in.
The truth itself has an independent existence, and transcends all other things.

The circle is a piece of abstract two dimensional geometry invented by humankind. It doesn't exist beyond our imagination. Certain objects in the observable universe might appear circular when projected onto a plane, and because we have invented circular geometry we are in a better position to measure certain aspects of those objects, but the circle itself is still an abstract. Circles allow us to describe things that have some circular properties, and we have arrived at the abstract notion of a circle through observing objects with circular properties.

starshine1234

Maths is not considered to be invented. It is considered to be discovered.

That is the difference. Maths seems to exist somehow on an independent level.

I agree that not everyone agrees with that.

Pythagoras didn't invent his theorem, he discovered it.


https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis
quote from Wiki page = Our external physical reality is a mathematical structure.

That is, the physical universe is mathematics in a well-defined sense, and "in those [worlds] complex enough to contain self-aware substructures [they] will subjectively perceive themselves as existing in a physically 'real' world"

end quote.

I don't understand it.

Kidchameleon

How can we know that Math transends universes when our one is the only one we yet know of? What if our "host" universe is four dimensional? What if PI is 7.14.... but our hosts have us beleive its 3.14...? What if triangles dont exist in the "real" universe? What if numbers dont exist? We are already seeing holes in our understanding when we consider things like entanglement. If / when humans create a simulated universe, we could make it entirely two dimensional. Who knows what crazy "math" our subjects would come up with.

smacl

starshine1234 wrote: »

Maths is not considered to be invented. It is considered to be discovered.

Considered by whom exactly? There is no single accepted definition of mathematics. If you study fields such as discrete methods, algorithmics, and combinatorial geometry you quickly realise there are multiple correct solutions to the same problem, and similarly multiple correct descriptions for the same geometry. For example, our circle could be described as the locus of a point travelling a fixed distance around an origin. We could also discretely consider it a polygon with an infinite number of sides. We could plot it as (r sin(a), r cos(a)) or by solving X^2+y^2 = r^2. Mathematicians are regularly inventing new and better ways to describe things, to meet evolving demands.

To be fair, the difference between discovery and invention in this context is entirely semantic.

starshine1234

I see what you mean.

However, if we met aliens we wouldn't expect them to have exact copies of Shakespeares plays, that they themselves wrote. But we would expect them to have the same mathematics as us.

That seems to suggest that a playwright 'invents' his plays, while a mathematican 'discovers' his theorems etc.

I agree it's a semantic argument but at the same time there does seem to be something more to it.

I'd go as far as to suggest that the maths will be the same in all universes, even in universes I can't conceive of. That certainly isn't true of Shakespeares plays.

So why does maths have a universal consistency?
Why is maths fundamental?
Why is our universe so perfectly described by maths? (alternatively, why is maths so good at describing our universe?)

smacl

starshine1234 wrote: »

However, if we met aliens we wouldn't expect them to have exact copies of Shakespeares plays, that they themselves wrote. But we would expect them to have the same mathematics as us.

Maybe not. We use a base 10 number system because we have 10 fingers, and use electronic digital computers because of the invention of the d-type flip-flop as the first high speed mechanism to store data persistently. Much of our mathematics now involves computability on the basis of Turing machines as a result. We use 2d and planar geometry a lot since we write stuff down on flat surfaces.

An alien that manipulated tools with a different number of fingers would not develop a base 10 number system. An alien with no fingers might not use discrete numbers at all, work solely with continuous ranges, and might develop analytical engines which are not based on discrete storage at all. This would all require and result in different maths. If we talk about different universes, you would get different maths based on different physical observations subject to different rules.

Mathematics is as infinite as imagination, and I could certainly imagine very different mathematics being developed to suit different contexts.

swampgas

At this point in this discussion I think Godel's Incompleteness Theorem should be invoked

https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

recedite

Anyone else now regretting they slept through so many maths classes?

smacl

Kidchameleon wrote: »

What if our "host" universe is four dimensional?

Our universe is often considered four dimensional, these being three cartesian dimensions plus time (x,y,z,t), or two spherical polar dimensions plus time and distance (Φ, θ,d,t) You can also tack on many extra dimensions to objects within our universe such as linear and angular velocities and accelerations, mass, volume, temperature or energy level etc..

starshine1234

swampgas wrote: »

At this point in this discussion I think Godel's Incompleteness Theorem should be invoked

https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

The implications of the incompleteness theorem is that maths can never be completed.

In other words, the maths system in use can always be extended, and new theorems can always be found.

That is not too different from saying that the english language will never be complete. It will always be possible to create new words.


Godels Theorem.
In any fairly advanced formal maths system, there are statements which can be made which can be seen by humans to be true, but which cannot be proven to be true within the formal maths system.

But, the formal maths system can be extended, by new theorems, and if you do so, the previously unprovable statement can now be proven.

However, by extending the maths formal system, you have now allowed for a new, different statement to be made which cannot be proven within the formal maths system, but which can be seen by humans to be true.

That process of extension, and further extension, can go on for ever.

The conclusion therefore seems to that there will never be a complete maths sytem, in which every true maths statement can be proven.


So, from where does this endless complexity arise?

The Godel theorem supports my argument that maths is endlessly complex, and that maths seems to have an existence of its own.



On the issue of number bases.
I did ask if the number pi would look different in base 6, for example. I realised afterwards that I was wrong to ask the question.

It is a simple mechanical operation to change the base of a number, and it cannot introduce new maths or new situations. The base in which a number is expressed has no effect on the mathematical qualities of the number.

A prime number is still prime for example, and an irrational number is still irrational.


I don't think aliens have access to different maths, even in different universes. Of course they may have developed different types of tchniques and things like that but that's only a trivial difference.


All Turing machines are equivelent. Some may be faster or more efficient at certain types of problems but all Turing machines are equivelent in that they can solve the same types of problems.

Quantum computers allow for new types of problems to be solved, or perhaps the correct phrasing is that they allow certain problems to be solved very quickly.
David Deutsch says that that proves that there are multiple universes. The only way huge numbers can be factored in a short time is if the calculation is farmed out to huge numbers of universes.
Pretty compelling argument I'd have thought.
HOw else do quantum computers perform their 'magic'?

smacl

starshine1234 wrote: »

However, by extending the maths formal system, you have now allowed for a new, different statement to be made which cannot be proven within the formal maths system, but which can be seen by humans to be true.

That process of extension, and further extension, can go on for ever.
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I don't think aliens have access to different maths, even in different universes. Of course they may have developed different types of techniques and things like that but that's only a trivial difference.

Consider those two statements, on the one hand you are saying that maths is infinite. Thus human maths at any point in time is a finite subset of all possible maths.

If you allow for an alien intelligence it would also have a knowledge of maths that is a subset of all possible maths.

On what grounds do you suppose our subset coincides or even intersects with the alien subset, particularly if we're talking about a different universe with a different set of observable physics?

starshine1234

I agree with that, but only to a degree.

I believe its the case that the more advanced maths is built upon the more basic maths.

Maths is infinitely extendable, but it might not be possible to develop some theories without first developing the necessary foundation for that theory.

So, some basic maths is essential as a foundation for the more advanced maths. I don't think the advanced maths can be developed without some essential basics.

I think the concepts of numbers is essential. The concept of unity, or oneness, the concept of twoness, the concept of threeness etc.

Apparently it's hard to get computers to understand the concept of 'threeness' for example. Computers can do amazing calculations but they don't understand that 3 means 3.

Humans apparently know what the concept of threeness means becuase we are grounded in the real world, and we have real experience of seeing things in threes.

At first there were only positive integers. No negative numbers, or real numbers, or the number 0, or irrational numbers, and certainly not complex numbers (square root of -1).

robindch

swampgas wrote: »

As software is inherently deterministic the only way to get true randomness is with external input.

Deterministic? P==NP?

smacl

robindch wrote: »

Deterministic? P==NP?

Ah, but does P==NP?

Edit: And for those of you with better math skills than I, there is a prize of $1 million to prove the above