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January 21st, 2019, 05:47 AM   #11
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This post addresses the OP, not JeffM1.

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Originally Posted by JeffM1 View Post
There is no physical possibility of representing exactly in decimal notation the fraction
But there is in other bases:
$$\frac13 = 0.1_3=0.2_6=0.3_9=\ldots$$
This shows that the "problem" (see below) is one of the decimal representation and nothing to do with the number itself. It also shows that we have representations that can present $\frac13$ exactly, and this is why mathematicians tend to use fractions such as $\frac13$ itself rather than decimals.

I wrote "problem" in quotes, because it isn't a problem. It's just a characteristic. Every representation has benefits and disadvantages. It's up to the user to select the one that works best for them in each given context. The problem is in your attempt to use the decimal representation for things that it isn't good at.

It's interesting that you say that "it is impossible to either count up or down from it". The fact is that it is impossible to either count up or down from it. That is a characteristic of the real numbers (not any particular representation) that confuses many students.

Finally, I need to point out that Mathematics is not Science. Science is "right" when it explains real world phenomena. Mathematics doesn't care about the real world. It is a tool that is used by scientists, but Mathematics is right when the logical deduction is accurate regardless of the real world.

Science doesn't have all the answers, but it is honest about those it doesn't have and why. Sometimes it's because we don't yet understand. Other times it's because the physical properties of the universe prevent us from getting answers.

Mathematics doesn't have all the answers, but again it's honest. It isn't about the real world anyway, but in its own world it is honest. We have proved (mathematically, logically) that there will always be things that can neither be proved nor disproved. And of course there are things that have just not yet been solved.

Religion doesn't have the answers either, but is not so honest about it and frequently tries to stop people asking the questions.
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January 21st, 2019, 05:49 AM   #12
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@Micrm@ss

OK point taken. Referencing my original post the stance from the sciences is often derisory to alternative opinions which probably influenced the tone of my opening gambit.

I've just found my way to duodecimal which at a glance seems to be an immediate step in the right direction.

Please explain why there is general ambivalence to a fundamentally flawed system practised by the best minds of humanity.
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January 21st, 2019, 05:55 AM   #13
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I find it strange that we base our whole mathematical system whereby decimal representation is mostly approximate.

Especially since this underpins many aspects of science and all 'proved' theorems.
This is all down to your misunderstanding of the difference between numbers and representations of numbers. I hope my previous post sheds some light on the matter.

In particular, mathematical and scientific theories do not rely on the decimal representation of numbers for their validity. Where approximations are used, they are understood and managed so as to not interfere with the validity of the conclusions.
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January 21st, 2019, 05:58 AM   #14
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Originally Posted by v8archie View Post
This post addresses the OP, not JeffM1.


Mathematics doesn't have all the answers, but again it's honest. It isn't about the real world anyway, but in its own world it is honest. We have proved (mathematically, logically) that there will always be things that can neither be proved nor disproved. And of course there are things that have just not yet been solved.

Religion doesn't have the answers either, but is not so honest about it and frequently tries to stop people asking the questions.
So once a representation no longer reflects the real world it is a representation best suited to its own rules and mechanisms.

So whilst 13=0.13=0.26=0.39=…(formula above) is correct according to Math the existence of God must also be correct according to the bible.

Unless I'm missing the point which is infinitely probable.

Last edited by skipjack; January 21st, 2019 at 01:02 PM.
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January 21st, 2019, 06:05 AM   #15
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So once a representation no longer reflects the real world it is a representation best suited to it's own rules and mechanisms.

So whilst 13=0.13=0.26=0.39=… is correct according to Math the existence of God must also be correct according to the bible.

Unless I'm missing the point which is infinitely probable.
Pff, yes. If you want to put it like this. That is correct.

Math works from a system of axioms, definitions and derivation rules. A very definite list. If you agree with all of those, then you must agree with the conclusions, that is a logical necessity.
This indeed corresponds to the bible. If you choose to accept the commandments from the bible, you must accept there is only one god. It's the same thing.

Math never says anything about the truth of axioms, definitions and derivation rules. It just says something about IF you accept them as true. Math never says it MUST be true.

Who says decimal expansions or even 2+2=4 is true? Because the axioms are true. Who says the axioms are true? Common experience tells us this. You can choose not to accept some axioms or derivation rules though. It is your right, and you'll get other conclusions than the rest of us. It is part of physics and even philosophy to defend the axioms of math, not of math.

There is one important distinction though, between math and religion. In math, every single step and dependence on axioms is carefully stated. If you want to prove 2+2=4 from basic axioms. Math outlines precisely how to do it and what axioms to use. Religion is much more vague in this respect.
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Last edited by skipjack; January 21st, 2019 at 01:05 PM.
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January 21st, 2019, 06:22 AM   #16
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So once a representation no longer reflects the real world it is a representation best suited to it's own rules and mechanisms.
To over-simplify, mathematics is about numbers, not representations of numbers. The truths about numbers that it reveals can be used by science to create theoretical models about how the real world works.

Representations of numbers are just filters through which we see the numbers themselves, just as photographers use filters to show the real world. Sometimes choosing a particular filter shows detail that is difficult or impossible with others, leading to new appreciation of the subject. They start from the observable truths to determine answers of "why", or from known truths, use logical deduction to find more truths.

On the subject of religion/God. The premise of Mathematics/Science is that everything must be proven from simpler bases of knowledge. Progress is achieved by demonstrating errors in previous thought and improving theories and models accordingly. The premise of religion is, generally, that the answers are already known and are correct. There is generally no progress, because there is resistance to highlighting errors in previous thought. It starts from "why" and attempts to work backwards to observable truths and thus can never really explain anything. There is no progress because the answers are already built into the system.

Besides, religion is not intended to be a system for understanding how the universe works. It's intended as a blueprint for interacting with others, codifying best-practice into societal norms and making the unknown less scary.
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Last edited by skipjack; January 21st, 2019 at 01:08 PM.
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January 21st, 2019, 06:28 AM   #17
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Thanks for all the contributions. This has given me some extremely valuable food for thought.

There can be no doubt that our current mathematical model has things ‘near as dammit’ from my own experience with structural calculations. I do just wonder though if we had the chance to forget everything we know and start from scratch; would we do anything differently.
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January 21st, 2019, 06:32 AM   #18
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Quote:
Originally Posted by sintan View Post
Thanks for all the contributions. This has given me some extremely valuable food for thought.

There can be no doubt that our current mathematical model has things ‘near as dammit’ from my own experience with structural calculations. I do just wonder though if we had the chance to forget everything we know and start from scratch; would we do anything differently.
If you would ask this to me "If you could start from scratch and redo math from the start, what would you do differently", there is going to be a whole LOT of things I'd do differently.

Math, like everything, is a cumulative process. Which means that often some person does something that seems logical to him, but 100 years later is a big source of confusion for the others.
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January 21st, 2019, 07:17 AM   #19
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Quote:
Originally Posted by sintan View Post
Hi Jeff and thank you for a reply free from sarcasm.

I find it strange that we base our whole mathematical system whereby decimal representation is mostly approximate.

Especially since this underpins many aspects of science and all 'proved' theorems.

I believe the whole representation of integers needs to be torn up and revisited.
This conclusion must have been reached by many and I was interested if anyone had experimented with an alternative system.

If someone tells me something is not perfect but it's the best they can come up with my natural instinct is to investigate a way to make it perfect.
Mathematics is based on imagining ideal worlds. We don't live in an ideal world. Therefore the mathematics of the ideal world does not exactly fit the real world.

We approximate pi as 3.1415926 when that is good enough for what cannot be shown to be a perfect circle.

Perfect triangles exist in the ideal world of Euclidean geometry and have, in that ideal world, angles whose measures sum to the sum of the measures of two right angles. If you measure the angles of a purported triangle and find that their sum is not 180 degrees, that means that (a) the purported triangle is not a perfect triangle, (b) our world is not the ideal world of Euclid, or (c) both.

If you are looking for an argument in favor of the divine and revealed knowledge, the fact that what we learn from imagining the ideal can be applied successfully though inexactly to the real world is not a bad argument. I do not expect the ideal to be perfectly manifest in the mundane.

You have a cake and three children. They ask you to divide it into three equal pieces. You can refuse and eat the entire cake yourself on the grounds that (a) it is impossible to ascertain whether the cake has a number of atoms evenly divisible by three, and (b) it is impossible to ensure that each piece of cake has exactly the same number of atoms even if the cake happens to have a number of atoms divisible by three. Alternatively, you can divide the cake into three approximately equal pieces.

In the imaginary world of mathematics, 1/3 is always an exact concept. In the real world of experience, sometimes 1/3 is a mere approximation.
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Last edited by skipjack; January 21st, 2019 at 01:11 PM.
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January 21st, 2019, 07:46 AM   #20
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Originally Posted by v8archie View Post
This post addresses the OP, not JeffM1.



But there is in other bases:
$$\frac13 = 0.1_3=0.2_6=0.3_9=\ldots$$
I was thinking this too as I read the previous posts. There are some reals where the expansion for them in any base does not terminate (e.g., $\pi$).

Of those reals that have no terminating expansion in any base, some are computable while others are not. By computable, I generally mean that we can calculate in a finite number of steps the decimal (or any) expansion of the number to any desired precision. It happens that $\pi$ is one of these numbers, which is why we can calculate out 3.14.... Further, no other real number can produce the decimal expansion of $\pi$, so where the expansion is unique to $\pi$, we can assert it is a valid representation of $\pi$ for our purposes.

The question of whether to work with only the computable reals as opposed to the 'regular' real numbers has been considered. You may feel more comfortable working with only those numbers that can be computed, but then you'll achieve results that differ from those derivable if working with the full set of reals.

It's worth noting that Kurt Gödel's work demonstrates that, given a consistent formal system capable of modeling basic arithmetic, there are true mathematical statements that cannot be proven within the system. If we revise the system so as to prove a particular statement that was unprovable in the former system, we simply end up with other true statements that are not provable (again, assuming the system is consistent). Take a non-computable real number as an example. We can enumerate all of the sentences of a formal language and never find one expressing a function capable of deriving the expansion of the noncomputable real number to any desired precision. We nevertheless know the particular noncomputable real number exists (as it can be shown to differ from each computable real in a listing), so we are left knowing there is a true statement that cannot be proven. Why do you think we would want to work with such statements that cannot be proven? Should we just assert they don't exist instead?
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