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Author: Subject: Our Entire Maths System is Fundamentally Wrong
aga
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[*] posted on 28-11-2016 at 14:37
Our Entire Maths System is Fundamentally Wrong


Mind (such as it is) provoked by blogger's Quantum thread, the whole Imaginary Number thing has been bothering me a lot.

The Arabic Number System that we use has been evolving for a few centuries, the first amazing Discovery being Zero.

Now, the square root of -1 had to be deemed 'an imaginary' number to make the maths still work.

OK. I can see how this is useful (thanks to the aforementioned thread) and how functioning mathematical results are obtained.

My main concern is that if the Number System in use Fails in any case, requiring any bodging, like <i>i</i>, then this indicates a fundamental error in the number system itself.

By itself <i>i</i> points to an illuminating Flaw in the counting system, probably in where + and - occur, and possibly the relationship between two numbers of equal magnitude with opposite sign.

Possibly +1 is not exactly equal in magnitude to -1, depending on the other parameters that lonely 1 has to experience to get from + to - or vice versa.

Edit:

The Correct response to finding such a flaw would to be fix the System, not apply a bodge to cover it up.

Despite the centuries of effort invested in a possibly incorrect system, Future progress could happen with an improved system.

Sticking with a sinking ship tends to get all the people and rats killed in the end, so it may be an area worth exploring.

[Edited on 28-11-2016 by aga]




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[*] posted on 28-11-2016 at 15:20


You can't say it's wrong as long as it actually works. Call them fudge factors if you like, but if they actually work (rather than providing approximations) then it must be right, at least in some sense.

Not that I understand IN, but it has been used for centuries and seems to have practical use. So somehow I doubt it's just a clever approximation that "kinda works" or math wizardry with no practical relevance to the world.




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[*] posted on 28-11-2016 at 15:32


Wow aga. You have said a lot. The bottom line is that there is nothing really wrong with the number system. The advanced answer is that if we ever find our current systems deficient in some way (or even uninteresting) then mathematicians have the license to change a few axioms and explore an entirely different system.


Mathematics quickly falls into two distinct areas -- traditionally labelled pure and applied.
Pure mathematics is built on axioms which IMO are boring little things. There exists an entity called a point. Every number has a successor. Given more than one item it is possible to choose between them. That kind of thing.
From these axioms our number systems (and other systems) are built. The concept of numbers having a successor leads to the concept of addition and the natural numbers. Repeated addition leads to multiplication and repeated multiplication leads to exponentiation.
Taking the inverse of these operations leads to subtraction, division, logarithms and roots. And these lead either to unanswerable questions or force us to step out of the natural number system. For example 3 subtract 7. Or 19 divided by 8. or square root of negative one. This is where the integers, rational numbers irrational numbers and complex numbers come from. (Real numbers (as distinct from irrationals) are a bit of a weird beast but that's another story.)

By the time you get to complex numbers you arrive at what is called a complete system. The basic operations that flow from the axioms always return a value that is within the complex number system. This is nice. This is why I like complex numbers.

Of course mathematicians don't always want to play within the rules. They add or remove axioms and see what happens. What happens if 0.99999999... is not equal to one? What happens beyond "infinity"? What if a×b is not equal to b×a? What happens if the thing you are interested in is represented by more than one number? These questions give rise to other useful systems such as the surreals, hyperreals, vector spaces, quaternions and also things such as topological spaces, manifolds and algebraic fields.

The thing is that all of these only exist in the mind of the mathematician. And the only rule is that they must be internally consistent. There are good reasons to stay within the realm of complex numbers -- just as there are often good reasons to restrict oneself to reals or natural numbers. But to state that mathematics is broken because sometimes we are forced to step from one system to another is incorrect.


And that brings us to the applied side of things. Mathematics is necessarily abstract. You might complain about i, but you overlook that all numbers are abstract. Show me a 2! Even the simplest things are abstract concepts.

What we do find is that mathematical ideas (including numbers) are useful. So, if we have questions about our world, a natural thing to do is to model things using mathematical systems. Sometimes in exploring the answers to those questions we stay within the confines of comfortable systems. Sometimes we don't. Sometimes we even have to invent systems that accurately model the thing we are interested in. (Ask me about toy train tracks some time.)

In applied mathematics we are really only interested in solutions that answer our real world questions. We might for example ignore negative or complex solutions to polynomial equations. But the pure mathematician will be provoked by such things and in the pursuit of internal consistency will explore those avenues and see what happens.




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[*] posted on 28-11-2016 at 16:18


Complex numbers (those that have some component of i in them) are actually more natural than just the 'real' numbers (with no i), at least in this sense:

If you make an arbitrary polynomial, p(z) = a + b*z + c*z^2 + d*z^3 + ... e*z^N

This only has all N roots if we consider z as complex numbers. If we limit ourselves to 'real' numbers, than sometimes there will be all N roots, but sometimes they would go missing. E.g. p(z) = 1 - z^2 has two real roots, but they would be missing for p(z) = 1 + z^2. So the 'imaginary' i makes the number set complete. You have similar issues with eigenvalues of matrices.

I think that the biggest problem is the word 'imaginary', which is kind of perjorative.


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[*] posted on 28-11-2016 at 16:26


Math is a good place to find interesting and deep questions and it sounds like you're orbiting a few. But remember the line from the movie Pi:

Quote:

As soon as you discard scientific rigor, you're no longer a mathematician, you're a numerologist!


"Imaginary" numbers aren't any more imaginary than "real" numbers, and I don't think that I'd even call them fudge factors. They're a conclusion of a generalized algebraic framework that has been made to describe what a number is and how they behave.

It would be true that formal inconsistencies would cause *severe* problems in math. But imaginary numbers don't introduce any inconsistencies so far as you have discussed. You can balance your checkbook all day long (ie, using a group defined on a subset of the rationals with the standard addition operator) without ever coming to a conclusion that complex math would disagree with. Complex math is built around familiar algebraic structures (the imaginary numbers are completely isomorphic to the real numbers), so it's hard to see where the contradiction would arise.

If you really don't like imaginary numbers, there's nothing stopping you from defining the square root function's domain to the non-negative integers only (indeed, many computers do this by default). Or you can redefine the function piecewise at zero, by defining sqrt(-1)=1. As long as you're internally consistent, this will be as legit as non-Euclidean geometry is. But, you might not get the algebraic structure out of your modified number system, which makes the standard number system powerful and useful.

If you're interested in the hows and whys of the development of complex math, this is a nice history IIRC:

Nahin, P.J.: An Imaginary Tale: The Story of √-1

If you really want a deeper dive into how math is built (also artificial intelligence, buddhism, and terrible puns)...
"Godel Escher Bach: An Eternal Golden Braid" - Douglas Hofstader
https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach

Ed: this one is just funny :D

Mathematical Cranks - Underwood Dudley

a couple things you said that stand out:

Quote: Originally posted by aga  
Now, the square root of -1 had to be deemed 'an imaginary' number to make the maths still work.


Don't get caught up in the name of a thing! If they were called Potato Numbers, it wouldn't mean they were tasty with ketchup. They are just another idea that was named by its detractors, like the Big Bang and Schrodinger's Cat.

Quote:
My main concern is that if the Number System in use Fails in any case, requiring any bodging, like <i>i</i>, then this indicates a fundamental error in the number system itself.


It's hard to see how. The idea of imaginary numbers arose from the building of a general mathematical structure, which unified the arithmetic of finance, for example, with the transformations of a Rubik's cube. Your argument sounds like you're throwing out electromagnetism, because early scientists of electricity didn't have the foresight to include magnetism in their studies.

Quote:

By itself <i>i</i> points to an illuminating Flaw in the counting system, probably in where + and - occur, and possibly the relationship between two numbers of equal magnitude with opposite sign.
Possibly +1 is not exactly equal in magnitude to -1, depending on the other parameters that lonely 1 has to experience to get from + to - or vice versa.


This... is an extremely bizarre remark for many reasons, not the least of which being no notion of magnitude or measurement is necessary to define negative numbers. They are defined by an algebraic structure called a group. All you need to build something that behaves like -1 is the natural numbers (0,1,2...) or certain subsets, and a very austere system of addition.






[Edited on 29-11-2016 by mayko]




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[*] posted on 28-11-2016 at 21:01


In the field of electronic and electrical engineering complex numbers are incredibly useful, as they can describe both phase and magnitude in AC systems (just one example). The beauty of it being this crazy math involving the square root of -1 actually produces real, useful results in real world systems.
I wouldn't call that broken!




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[*] posted on 28-11-2016 at 21:43


A topic on math? On my favorite chemistry site? AND it's about complex numbers? It's like a dream come true :D

Props to mayko, pantone159, and j_sum1 for defending my mistress i

just as a side note, this video series does a great job of discussing this topic with beautiful visuals:
https://www.youtube.com/watch?v=T647CGsuOVU
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[*] posted on 29-11-2016 at 00:41


Quote: Originally posted by JnPS  
A topic on math? On my favorite chemistry site? AND it's about complex numbers? It's like a dream come true :D

Your luck is very good indeed ! Try these :-

http://www.sciencemadness.org/talk/viewthread.php?tid=62973
http://www.sciencemadness.org/talk/viewthread.php?tid=65532

My instinct says there's something amiss with the concept of Minus generally.

I'm not clever/educated enough to produce algebraic arguments and proofs either way.

Where it feels wrong is that physically we can only really justify 0, and not 0.

Rewind back to the beginning of maths-time.

Ug has 3 apples, Zog only 1, i.e. U=3, Z=1

Zog eats his apple, and now has 0 apples. U=3, Z=0

He then gets hungry and begs to borrow an apple from Ug.

Ug agrees and lends an apple to Zog, so now U=2, Z=1

Zog then eats his apple. U=2, Z=0

Where we go from here requires the addition of a whole other universe almost completely unrelated to the absence or presence of any arbitrary number of apples.

If Ug is huge, powerful and insistent, or Zog is honourable, we can write U=3, Z=-1

If Zog stubbornly refuses to return any apples, ever, it splits:

Ug's version : U=2+1, Z=-1
Zog's version: U=2, Z=0

In reality, Zog's version is correct <i>at this time</i> as there physically exist only 2 apples in the system.

Ug has invoked a non-existant Anti-apple to represent the position and magnitude of a number of real apples <i>at some future time</i>.

It is at best a Prediction dependant on unreliable assumptions : Zog's attitude and the likelihood that new apples will ever come, neither of which are accounted for in the equations.

Ug may physically have 2 or 3 apples in the future, despite the pure maths being undeniably correct in both versions.




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[*] posted on 29-11-2016 at 01:05


Aga, you scare me sometimes because I've had very similar thoughts about almost the same thing. I get stuck with 1+1 =2 as in an apple + apple = something new (2). I can understand 1+1= 2(1) or 2 of 1. Similarly when 1 - 1 = 0 where zero can be considered impossible in some cultures.

At nights when drifting off I get these glimpses of strange things like this and deeper insight that often is totally elusive upon waking.

Quote: Originally posted by aga  
Quote: Originally posted by JnPS  
A topic on math? On my favorite chemistry site? AND it's about complex numbers? It's like a dream come true :D

Your luck is very good indeed ! Try these :-

http://www.sciencemadness.org/talk/viewthread.php?tid=62973
http://www.sciencemadness.org/talk/viewthread.php?tid=65532

My instinct says there's something amiss with the concept of Minus generally.

I'm not clever/educated enough to produce algebraic arguments and proofs either way.

Where it feels wrong is that physically we can only really justify 0, and not 0.

Rewind back to the beginning of maths-time.

Ug has 3 apples, Zog only 1, i.e. U=3, Z=1

Zog eats his apple, and now has 0 apples. U=3, Z=0

He then gets hungry and begs to borrow an apple from Ug.

Ug agrees and lends an apple to Zog, so now U=2, Z=1

Zog then eats his apple. U=2, Z=0

Where we go from here requires the addition of a whole other universe almost completely unrelated to the absence or presence of any arbitrary number of apples.

If Ug is huge, powerful and insistent, or Zog is honourable, we can write U=3, Z=-1

If Zog stubbornly refuses to return any apples, ever, it splits:

Ug's version : U=2+1, Z=-1
Zog's version: U=2, Z=0

In reality, Zog's version is correct <i>at this time</i> as there physically exist only 2 apples in the system.

Ug has invoked a non-existant Anti-apple to represent the position and magnitude of a number of real apples <i>at some future time</i>.

It is at best a Prediction dependant on unreliable assumptions : Zog's attitude and the likelihood that new apples will ever come, neither of which are accounted for in the equations.

Ug may physically have 2 or 3 apples in the future, despite the pure maths being undeniably correct in both versions.
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[*] posted on 29-11-2016 at 05:21


Complex numbers are a tool, they can describe a plane, or a signal with phase and as a tool it works.

For more of a head trip see https://en.wikipedia.org/wiki/Quaternion
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[*] posted on 29-11-2016 at 06:43


The introduction of the number i may look like an arbitrary thing to make solving certain equations possible. The fact, however, is that this is NOT arbitrary at all. Later in time, more than a century after complex numbers were introduced, the concepts of groups, rings and fields were developed over time and only fairly recently (150 years??) there is full understanding of these concepts.
Nowadays we have the branch of mathematics, called algebra, which is about the structure of sets, endowed with certain operators. It can be very abstract, but a very common set with a well known algebra is the set of real numbers R, endowed with the operators + (addition) and x (multiplication). The set R with its operators (+, x) is called a field. It appears that equations, fully specified in terms of this field, do not necessarily have solutions in this field. An example is the equation y*y + 1 = 0. This equation can be described only in terms of basic operators (x and +) and real numbers, which are in R. Yet, there is no solution in R. What we now know from algebra is that such sets can be extended by introducing a formal number, outside of the field, with a property, which no number inside the field has. For the set R, with operators + and x, the introduction of a number with the property i*i+1=0 extends the field R, such that the above equation y*y+1=0 can be solved. In fact, using this extension, ANY equation over R can be solved with either elements in R, or with elements of the form A+Bi, with A and B in R.

This theory can be expanded further. Another interesting field is the field of quotients Q, which consists of number A/B, with A and B integers. Suppose we want to solve the equation y*y*y-2=0 in Q. This is not possible. There is a solution in R, but not in Q. We, however, can extend Q with a special number w, which has the property w³=2. If we introduce this number, then we get numbers of the form A+Bw+Cw² with A, B, C elements of Q. The nice result is that any "number" of the form A+Bw+Cw², when added to, multiplied with, divided by or subtracted from another number of the same form again is a number in this same set. This set is denoted Q(w) and Q(w), with operators + and x again is a field.

The field of complex numbers R(i) is the only possible field extension over R, and usually is written as C. Any other field extension over R either equals R, or equals C.

What initially may look like using a flawed number system, which is bodged to get it to work a little better, in reality now is part of a deep understanding of algebras and number systems.

If you really want to know more of this fascinating subject, then read about field extensions: https://en.wikipedia.org/wiki/Field_extension

More insight can be obtained by reading about Galois theory: https://en.wikipedia.org/wiki/Galois_theory

Interesting stuff, but do not expect an easy ride if your mathematical basic understanding is not very deep.

The subject of Quaternions, introduced by Marvin, is an interesting one as well, but these are not a field extension of R. Quaternions have the property that AxB is not always equal to BxA and hence they do not form a number system like the real numbers or complex numbers.




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[*] posted on 29-11-2016 at 12:25


Phew ! My faith is a little restored - thanks woelen.

The notion that a specific mathematical system applies Only to a specific set of circumstances sounds entirely reasonable.

Regarding the whole problem of Minus, here's a puzzle i heard years ago :-

A Hotel owner in a small town rents his rooms for $10 a night.

Guests must pay on arrival, although they have a 100% money back guarantee if they are not satisfied.

The hotel is empty, and only One guest is coming this week, so the Hotellier gets the local carpenter to fix a bedroom door, but he has no money yet, so owes the Carpenter $10.

The Carpenter hoped to get paid, but still needed to pay the Baker $10 for the bread he wanted that day, so he owes the baker $10.

The Baker's daughter had to stay in the hotel one night, but without the money from the Carpenter, he could not pay, so he owes the hotel owner $10.

They are anxious about the fact that they all owe $10 that they cannot pay.

The Guest arrives, and hands over the $10 to the Hotellier and goes to bed.

The Hotellier immediately runs out and pays the Carpenter, clearing his debt.

The Carpenter does the same and pays the Baker, clearing his debt.

The Baker goes next door and hands over his $10 to the Hotellier, clearing his debt.

Everyone is much happier, now they have no debts to pay.

In the morning the Guest looks dishevelled, and is clearly not happy and says he had a terrible night's sleep.

He demands his money back.

Reluctantly the Hotellier refunds him the $10 as promised.

The Guest leaves with his $10, never to return.


At the start we have a total debt burden of $300 for the three people.
$10 is introduced, then $10 is removed.
The $300 debt burden evaporates to $0 during this magical process.




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[*] posted on 29-11-2016 at 13:22


So now you see that money has no intrinsic worth, and is merely a placeholder by agreement.



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[*] posted on 29-11-2016 at 13:33


There is nothing at all puzzling about it (and you mean $30 not $300).

Money is a medium of exchange for goods and services, not something physical and concrete like a potato.

It is actually a nice little illustration of how economies work and the story was actually a common situation in the days of hard currency, when an inadequate specie supply could leave people with no money to pay other with. This is why paper money was created, so that the money supply could be grown to meet the needs of the economy.

Benjamin Franklin for example ran a newspaper with a shop attached. Why? Because there little hard currency in the New World (they had to ship what they had to Britain to pay for British goods) and he was often paid in things like tobacco or furniture which he then had to sell. He was a strong advocate for paper currency as a result.

The Carpenter could have paid the Baker with a chair. The Baker's daughter could have paid the Hotelier with loaves of bread, working as a scullery maid, or in some other way (ahem), but they all chose to keep accounts of debts instead (which is how international banking came into existence).

Or the Hotelier could have paid the Carpenter with a script good for one night at the hotel (basically creating private currency), which the Carpenter could have passed on to the Baker as payment, which would then have ended back in the hands of the Hotelier. No debts left, no money involved.

All the guest did was introduce the necessary liquidity for the others to settle their debts using currency, which they could have settled without it if they have not insisted on exchanging pieces of paper marked "currency" or coin (if it was hard currency).

When the Hotelier refunds the guest he is taking a loss, according to the terms he advertised.




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[*] posted on 29-11-2016 at 13:44


Quote: Originally posted by aga  

Regarding the whole problem of Minus, here's a puzzle i heard years ago :-
...The $300 debt burden evaporates to $0 during this magical process.


That's because each person has a $10 debt and $10 owed to them. Net worth zero before and after.




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[*] posted on 29-11-2016 at 14:30


Oops. Yes, i meant $30 in total debt.

The thing is, depending on the point in time you choose, the debt of each person changes.

i.e. the Baker starts off with -$10 if his debt is represented that way, or more realistically a +$10 debt as he will know and 'feel' that his debt is positively 10.

Nett zero for the overall time period, for sure, but Not nett zero between each step.

Time Dependance appears to be rather important, yet Time does not appear in the maths : total debt = d1+d2+d3

This ignores the external factors that affect the Positive value of each 'd' at each particular time, such as the passage of 1000 years, the disappearance of the debtor, or their death.

Overall, a Minus value (in this context) is meaningless, as it will equal Zero if given a large enough time window to become so.

There are still no Anti-apples, so a Minus value still has no actual physical reality, hence the notion of a Negative number is also a 'bodge' to try to make the numbering system work.

Edit:

Kelvin possibly had thoughts like these.
-273 C or -460 F make no sense.

Zero K works better, as it is truly the Zero point beyond which the Real world does not go.

[Edited on 29-11-2016 by aga]

[Edited on 29-11-2016 by aga]




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[*] posted on 29-11-2016 at 14:55


There was no net debt among the three people, even at the beginning, provable by simply having the three of them pass around a piece of paper written up by the Hotelier himself.

They could have zeroed out the accounts directly, if they had been recorded together in a ledger, without passing around anything. Organizations do this sort of thing routinely (and again, this is how international banking got started).

Lets cut out the third person.

If I do $100 work for you, but you do not pay me, and then I do $100 work for you, and you do not pay me, are we collectively $200 in debt, or zero in debt?

Normally people would agree to cancel the debts directly. In reality the net debt is zero. Similarly I can create a trillion dollars in fictitious debt by declaring that you owe me half a trillion, and I owe you half a trillion. It is just an accounting trick.




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[*] posted on 29-11-2016 at 15:18


Quote: Originally posted by careysub  
I owe you half a trillion.

Now i have it in writing ... could you please send the half a trillion Nitrogen molecules before the end of the month.

My point is that the Human Attitude affects the Maths chosen to represent Reality.

In doing so, i feel that something gets broken in favour of the understanding of Maths rather than in favour of understanding Reality.

If there is no possibility of a Real Negative, then where exactly are we ?

Side-tracked and Obsessed with our mental abilities that simply do not stack up well alongside those outside of our cosy nest-world.

The Acid Tests of human understanding are simple :

1. make one of You from scratch.
2. put yourself 100 light years from here, alive, now.
3. do not die for at least 200 years.

4. (optional) come back in 200 years, from 100 light years away, to here, yesterday, and tell me what it will-woulding-have-had-been like, and how much of a bastard i will be today for suggesting it, purely due to the dreadful grammar problems.




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[*] posted on 29-11-2016 at 15:24


Quote: Originally posted by aga  

There are still no Anti-apples, so a Minus value still has no actual physical reality, hence the notion of a Negative number is also a 'bodge' to try to make the numbering system work.

Edit:

Kelvin possibly had thoughts like these.

Zero K works better, as it is truly the Zero point beyond which the Real world does not go.


Possibly you do not believe in flight (or holes).

Be it a bird or a ball, it is something with altitude, i.e. something "above" the ground. What does this mean? Well we measure a distance (in some chosen unit) from the surface of the ground to the thing in flight. As it descends its altitude gets smaller and smaller, approaching zero.

But what's this? There is a well!

And the bird/ball descends into the well, below ground level. What is its altitude now? Well if we don't believe that negative numbers are "real"and thus there is no such thing as negative altitude then we must redefine our whole concept of "altitude" to be relative to the bottom of the deepest hole. So we must know the depth of the deepest hole in the entire world before we can say what the altitude of anything is.

Similarly if negative numbers do not exist as real things, then debts do not exist either. That's a relief!

It is an illusion to think that even the counting numbers are real things. They are convenient tools that humans invented. We have a sheep and another sheep. Is that "two"? No, its just a sheep and another sheep. "Two" is an invented abstraction. Some primitive cultures do not have these invented abstractions, and exchanges of goods are not done by the artificial system of "counting" them, but by pairing them up.

EDIT: (Pairing up small ratios, what we would call two-to-one, or three-to-one, is done without counting them simply by direct observation. Mammals (dogs for example) and many birds, are able to distinguish small numbers of objects in this way without a symbolic number system.

[Edited on 30-11-2016 by careysub]




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aga
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[*] posted on 29-11-2016 at 15:39


Altitude above some arbitrary ground is meaningless, unless you're an earth-bound pilot.

It's distance from the centre(s) of the largest gravitational entities that makes most sense if you're High enough.




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[*] posted on 29-11-2016 at 16:42


Quote: Originally posted by aga  
Altitude above some arbitrary ground is meaningless, unless you're an earth-bound pilot.

It's distance from the centre(s) of the largest gravitational entities that makes most sense if you're High enough.


So the concept of flight could not exist until we knew the size and shape of the Earth?

All pilots and birds would disagree that the altitude above the ground is meaningless.




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[*] posted on 29-11-2016 at 19:29


Quote: Originally posted by aga  
... that makes most sense if you're High enough.


And that pretty much sums up aga's posts.....




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[*] posted on 29-11-2016 at 20:14


Quote: Originally posted by aga  
There are still no Anti-apples, so a Minus value still has no actual physical reality, hence the notion of a Negative number is also a 'bodge' to try to make the numbering system work.


Kronecker once said, ""God made the integers [sometimes translated, the natural numbers], all else is the work of man," but he was a number theorist so it's a rather self-congratulatory remark. ;)

You are right that I can't show you a negative apple, but then again, you can't actually show me the number three. You might hand me some apples, or hold up some fingers, or point to a small pile of pebbles, but in no case have you shown me a three. Three-ness is an abstract property which is shared by certain collections of apples, fingers, or pebbles.

Does this mean that the number three isn't "real"?

Quote:

Morpheus: What is real? How do you define 'real'? If you're talking about what you can feel, what you can smell, what you can taste and see, then 'real' is simply electrical signals interpreted by your brain.


I tend to believe this: numbers are real, because the behavior of physical entities (such as mathematicians) depend upon their properties.


Quote:

Kelvin possibly had thoughts like these.
-273 C or -460 F make no sense.

Zero K works better, as it is truly the Zero point beyond which the Real world does not go.


This is not actually true, though the counterexamples are exotic! Negative absolute temperatures can exist for out-of-equilibrium systems with a small number of energy states:
https://en.wikipedia.org/wiki/Negative_temperature





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[*] posted on 10-12-2016 at 12:22


Mind further boggled : Even Zero is now causing problems.

$$0 - 1 = -1$$
ok. get that. Movement beyond a point called '0' in the 'down' direction. So far so good.
$$\frac11=1$$ $$\frac nn=1$$
Erm, yes, that also seems fine.
$$\frac00=0$$
$$\frac \infty \infty = 1, 0, \infty$$
WTF ?!?

Then there's some craziness called Riemann's paradox saying that:
$$\infty-\infty=\pi$$

Forgetting 'infinity' as that's just Nuts anyway, if n/n=1 then 0/0 cannot equal 0.

Either that or assumptions about the number series are basically flawed, or mathematical operations <i>whose proofs are based on real number sequences</i> are also flawed.

Probably it's all Fine and Dandy, just that i don't have enough brain capacity to understand how f(X)=1 can be fine for absolutely every number apart from 0. Or infinity. Maybe not SQRT(-1) either. Possibly not e<sup>-pi</sup>.

If it works, always, it is a Real Law, and f(X) does equal 1.

If it fails sometimes, it's got to be wrong (or at least something is).




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[*] posted on 10-12-2016 at 14:21


The Riemanns paradox is actually fairly simple (even I understand it). It only applies to certain infinite convergent series and simply states that you cannot rearrange such series without changing them.



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