Math, Assumptions, and the Supernatural

[Polymath257]

I am unfamiliar with some of your terms, Poly. Could you please elaborate on how you are defining "strong" here. As in "a strong assumption". Do you simply mean a strong assumption assumes more (and is less evidence based) than a weak assumption? Or is there more to it than that? In other words, how does one measure the strength of an assumption?

The strength of an axiom (or axiom system) is determined by how much can be proved from it.

So, the axioms for ordinary arithmetic (addition, multiplication, induction) are usually considered to be very weak since they say nothing about geometry (for example), nothing directly about set theory (and very little indirectly), etc.

On the other hand, the axioms of set theory allows us to talk about arithmetic, geometry, and set theory all in one set of axioms.

Also, if you look at set theory without the axiom of choice versus set theory *with* it, there are many situations where the theory without can prove limited results in a few cases while the theory with can prove much more general results in a wide range of cases.

This, then, gets into the question of how axioms are chosen. No longer is 'intuitively obvious' considered to be a criterion for our assumptions. The standard axioms for set theory are far from being 'intuitively obvious', but are almost universally accepted among mathematicians. But there are *extensions*, further assumptions that are known to be independent, that are either not universally accepted or are so technical that nobody really cares about them.

And what I have found is that we have certain *results* we want out of any axiom system. We want to be able to add and multiply numbers. We want to be able to find volumes, areas, etc. For modern math, we also want to be able to solve differential equations.

Then, we *choose* our axioms so that these minimal results can be achieved and the other results are as 'beautiful' as possible.

So, in a sense, beauty determines truth in math.

 

[Polymath257]

Why have I not been informed of this book before? I demand an investigation!

Yes, that is clearly a crime of humongous proportions!

It is a bit outdated in it's analysis of AI, but it is a fun read with lots and lots of wonderful questions and insights.

 

[Polymath257]

I like the term, "pustulate", but what is it?

Thanks for pointing that out. Fixed (along with a couple other typos).

As for making assumptions, it all boils down to usefulness.
We can construct all sorts of systems. I like the ones that
can model reality. Others can be fun too, eg, modular
arithmetic used in cryptography.

Usefulness is clearly one criterion. Especially when math is used as a language for other areas, this is important.

But internal to mathematics, the most often used criterion is that of aesthetics. Are the proofs nice and clear? Do we get those useful results quickly and easily? And what happens to the overall structure? is it 'clunkly', ad hoc, and rather arbitrary in use, or does it reveal hidden beauties and patterns that inspire more math?

 

[Revoltingest]

Thanks for pointing that out. Fixed (along with a couple other typos).



Usefulness is clearly one criterion. Especially when math is used as a language for other areas, this is important.

But internal to mathematics, the most often used criterion is that of aesthetics. Are the proofs nice and clear? Do we get those useful results quickly and easily? And what happens to the overall structure? is it 'clunkly', ad hoc, and rather arbitrary in use, or does it reveal hidden beauties and patterns that inspire more math?

Oh, you.....always obsessed with beauty.
(With my looks, it must be about my winning personality.)

 

[beenherebeforeagain]

Does mathematics exist outside of the mind that appreciates it? Does it have existence in some world of abstract ideas, (edit: sorry, that should have been ideals) separate from our mundane existence? (If yes, then that is supernatural, at least in my opinion...)

Or is mathematics something that is really only in the mind? (that is, it is an emergent property of mind, maybe? Or just a convenient invention?)

Or is there some other alternative?

 

[SoyLeche]

Does mathematics exist outside of the mind that appreciates it? Does it have existence in some world of abstract ideas, separate from our mundane existence? (If yes, then that is supernatural, at least in my opinion...)

Or is mathematics something that is really only in the mind? (that is, it is an emergent property of mind, maybe? Or just a convenient invention?)

Or is there some other alternative?

I'm not very familiar with it - but I think there is a Plato vs Aristotle argument about that topic.

 

[beenherebeforeagain]

I'm not very familiar with it - but I think there is a Plato vs Aristotle argument about that topic.

a debate that is still going on today--is math invented or discovered? There's interesting arguments on both sides...but as I understand it (or at least to my level of comprehension), they rely on different unproveable assumptions...

 

[SoyLeche]

a debate that is still going on today--is math invented or discovered? There's interesting arguments on both sides...but as I understand it (or at least to my level of comprehension), they rely on different unproveable assumptions...

I first was introduced to the philosophical argument in the book Anathem. It took a while after that to realize it was a Plato v Aristotle thing.

 

[Skwim]

Then comes the issue of religion.

People have many different views of what things are true in their religions. There are
multitudes of different religions (especially thorugh time) with huge variations on what
was believed, thought important, etc.

What if the existence of a supernatural is simply inconsistent?

In what way would an existence be inconsistent? With itself or with something else?

What if the existence of a supernatural is neither provable nor disprovable from our other
assumptions?

It would remain just that, an unknown.

What if the question really doesn't have a truth value at all?

It would remain just that---not being true or false---in limbo.

Now, if this is the case, what are the advantages and disadvantages of the assumptions either way?

What assumptions?

.

 

[halbhh]

I'm not sure how this analysis will go over in this forum, but here goes.

First, some background. Mathematics is proof based. The original example
of how math is done is Euclid's Elements, the first geometry 'textbook'.
What Euclid did was give a number of axioms and postulates: basic assumptions
of how things work that he considered to be 'intuitively obvious'. he then
proceeded to use those assumptions to prove various things about geometry,
numbers, etc.

But there was one postulate, the parallel postulate, that just didn't seem to
be as 'intuitively obvious' as the rest of Euclid's assumptions. Many people
over many centuries tried to prove this postulate from the other assumptions,
but failed to do so. So, while nobody actually believed the parallel postulate
to be false, nobody was able to prove it either.

Then, about 200 years ago, people started working with a system of geometry
that assumed the parallel postulate was false. And, what they eventually found
is that geometry with the parallel postulate and geometry with its negation
are *equally* consistent internally. In fact, using either, you can construct
a 'model' where the assumptions of the other hold.

This shook the foundations of mathematics. Math had always been held as the
most certain area of study. It was used by Plato as a crucial example showing the
existence of his 'Forms'. It had inspired Descartes to re-examine his assumptions,
leading to his break with Aristotelian philosophy.

But now, there were *two* equally consistent theories of geometry. Neither could
be discarded solely on the basis of logic. They were equally consistent internally.

One aspect of this revolution was that mathematicians started looking much closer
at the basic assumptions of *all* of math, even those underlying basic arithmetic.

One goal was to find a collection of assumptions that is consistent--where it could
be *proved* that no inconsistencies would ever be derived. It was also a goal that
it could be proved that these assumptions could answer any (mathematical) question,
this being called 'completeness'. It was felt that, even if we didn't have all the
'correct' axioms, there should be a collection of axioms that is both consistent
and complete and finding that collection of assumptions became a central goal for
many mathematicians in the early 1900's.

Then, a mathematician named Kurt Godel came along. He *proved* that any sysem of
axioms that is unambiguous (in a well defined sense) could NOT be both consistent
and complete if it was strong enough to deal with ordinary arithmetic.

So, the problem that had been raised by the parallel postulate in geometry was shown
to exist even in ordinary, elementary school arithmetic: it is simply not possible
to even prove arithmetic is consistent without making assumptions stronger than
arithmetic itself.

Moreover, any system of axioms strong enough to deal with arithmetic *must* have
questions it cannot answer. If consistent, it simply cannot be complete (and unambiguous
in the sense above). There MUST be questions that are unanswerble without making
additional assumptions. We say that such questions are independent of our axioms

And we know of many independent questions in mathematics: questions that simply
cannot be answered one way or the other from the axioms we use for modern math.
Some have been subsequently assumed (the Axiom of Choice). Others are still being hotly
debated (Continuum Hypothesis). Others are obscure even to most mathematicians (Martin's
axiom).

For ALL of these, we get different theories of *mathematics* by either assuming them
OR assuming them to be false. And *both* ways are equally consistent internally.

In a sense, these questions have no actual truth value, even though they are meaningful
and some are even important. You can use either the statement or its negation equally
well and have a mathematics that is equally consistent either way.

Bummer. Well, sort of.

Now, I have been pondering. We all make assumptions about the 'real world'. Among those
is that we are not the only conscious being, that there is an external world that our
senses can perceive, however imperfectly, that our memories are reliable to at least
some extent, etc.

We use these assumptions to move around in the world, learn how that world works, etc.
We develop science and technology, etc. We make art, music, and love others.

Then comes the issue of religion.

People have many different views of what things are true in their religions. There are
multitudes of different religions (especially through time) with huge variations on what
was believed, thought important, etc.

What if the existence of a supernatural is simply inconsistent?

What if the existence of a supernatural is neither provable nor disprovable from our other
assumptions? What if the question really doesn't have a truth value at all?

Now, if this is the case, what are the advantages and disadvantages of the assumptions either way?

To me any idea about God would be only (ever, always) just a conjecture or speculation.

but to experience something (something happens) is an event -- or fact -- something that occurs. To me God exists I know by events.

Regardless of how individuals, ideas, think on God In other words, while I don't object to all the views, theories, languages, ideologies in and of themselves, I also don't believe in them as....conclusive in a meaningful way.

Of course, they don't have to be poorly corresponding to events. General Relativity for instance corresponds to events in a high degree, so that it is valuable.

But in the end I really just I believe in what happens.

Events from the rest of the Universe which I experience.

Like sunlight, water, encounters, and much more.

 

[Dawnofhope]

Interesting post!

Then comes the issue of religion.

People have many different views of what things are true in their religions. There are
multitudes of different religions (especially through time) with huge variations on what
was believed, thought important, etc.

While there are many variations of religious beliefs, there is also a remarkable degree of consistency too. An estimated 55% of the world’s population identify as either Christian or Muslim. That percentage is predicted to rise to over 60% in the next 50 years. This increase despite the rapid development of technology and science. Then Hinduism and Buddhism account for another 20 - 25% of the world’s population.

What if the existence of a supernatural is simply inconsistent?

That depends on the assumptions we have about the supernatural. Not all theologies are equal from a rational or evidenced based perspective. What if a fundamental theological principle were the harmony of science and religion giving science the benefit of the doubt if the evidence was strong enough? So if science categorically disproves a young earth or a world wide flood, then abandon those theologies for something more plausible, consistent with science?

What if the existence of a supernatural is neither provable nor disprovable from our other
assumptions?

It depends on the assumptions.

What if the question really doesn't have a truth value at all?

Then why waste time considering questions of religious truth? There would be no point.

Now, if this is the case, what are the advantages and disadvantages of the assumptions either way?

One assumption could be the non-existence of a supernatural realm with god or Gods. The opposing assumption is the existence of god or Gods. The advantages or disadvantages depends on whether you’re a pragmatist or an idealist. For example an atheist perspective may theoretically have a stronger evidence base yet theism may empower someone to live a more noble life despite the irreconcilable contradictions.

Anyway, thanks for an interesting and thought provoking post.

 

[Sunstone]

a debate that is still going on today--is math invented or discovered? There's interesting arguments on both sides...but as I understand it (or at least to my level of comprehension), they rely on different unproveable assumptions...

Last I heard, there is some evidence that counting (arithmetic) is to some extent not only genetic-based behavior in humans, but also genetic-based behavior in some other animals. For instance, eagles can apparently count as high as three.

 

[blü 2]

Then, a mathematician named Kurt Godel came along. He *proved* that any sysem of
axioms that is unambiguous (in a well defined sense) could NOT be both consistent and complete if it was strong enough to deal with ordinary arithmetic.

How can we construct a parallel with religious belief from that?

Off the top of my head, the nearest thing to axioms in supernatural religious belief may be:

a) a supernatural world exists external to the self but not in nature
(b) in that world are beings who can affect reality magically (that is, independently of the rules of physics)
(c) there are known methods for keeping certain of those beings on your side, starting with worship.
(d) the only way we can and do know about this world is through the words of human authorities (texts, gurus, traditions &)
(e) the problem of determining which possible source is an authority is solved by faith, there being no objective test.

But whether or not that's right, if we don't know what the religious axioms are, we don't really have a place to stand when it comes to comparing them, do we?

And if it's roughly right, then we can see that whereas mathematics, although wholly conceptual, has rules for consistency, then one of the things about supernatural religion, whose supernatural aspect is also wholly conceptual, is that it doesn't.

 

[Polymath257]

How can we construct a parallel with religious belief from that?

Off the top of my head, the nearest thing to axioms in supernatural religious belief may be:

a) a supernatural world exists external to the self but not in nature
(

I think we can start with this one. I think this is likely to be independent of the previous assumptions. The failure of all attempts to prove it or disprove it points in that direction.

And maybe this is what 'faith' is: an acknowledgement of an assumption that is independent of the others.

 

[Ponder This]

What if the existence of a supernatural is neither provable nor disprovable from our other
assumptions? What if the question really doesn't have a truth value at all?

What you are describing is related to the agnostic point of view: that nothing can be known, claiming neither faith nor disbelief.

 

[Samantha Rinne]

I'm not sure how this analysis will go over in this forum, but here goes.

First, some background. Mathematics is proof based. The original example
of how math is done is Euclid's Elements, the first geometry 'textbook'.
What Euclid did was give a number of axioms and postulates: basic assumptions
of how things work that he considered to be 'intuitively obvious'. he then
proceeded to use those assumptions to prove various things about geometry,
numbers, etc.

But there was one postulate, the parallel postulate, that just didn't seem to
be as 'intuitively obvious' as the rest of Euclid's assumptions. Many people
over many centuries tried to prove this postulate from the other assumptions,
but failed to do so. So, while nobody actually believed the parallel postulate
to be false, nobody was able to prove it either.

Then, about 200 years ago, people started working with a system of geometry
that assumed the parallel postulate was false. And, what they eventually found
is that geometry with the parallel postulate and geometry with its negation
are *equally* consistent internally. In fact, using either, you can construct
a 'model' where the assumptions of the other hold.

This shook the foundations of mathematics. Math had always been held as the
most certain area of study. It was used by Plato as a crucial example showing the
existence of his 'Forms'. It had inspired Descartes to re-examine his assumptions,
leading to his break with Aristotelian philosophy.

But now, there were *two* equally consistent theories of geometry. Neither could
be discarded solely on the basis of logic. They were equally consistent internally.

One aspect of this revolution was that mathematicians started looking much closer
at the basic assumptions of *all* of math, even those underlying basic arithmetic.

One goal was to find a collection of assumptions that is consistent--where it could
be *proved* that no inconsistencies would ever be derived. It was also a goal that
it could be proved that these assumptions could answer any (mathematical) question,
this being called 'completeness'. It was felt that, even if we didn't have all the
'correct' axioms, there should be a collection of axioms that is both consistent
and complete and finding that collection of assumptions became a central goal for
many mathematicians in the early 1900's.

Then, a mathematician named Kurt Godel came along. He *proved* that any sysem of
axioms that is unambiguous (in a well defined sense) could NOT be both consistent
and complete if it was strong enough to deal with ordinary arithmetic.

So, the problem that had been raised by the parallel postulate in geometry was shown
to exist even in ordinary, elementary school arithmetic: it is simply not possible
to even prove arithmetic is consistent without making assumptions stronger than
arithmetic itself.

Moreover, any system of axioms strong enough to deal with arithmetic *must* have
questions it cannot answer. If consistent, it simply cannot be complete (and unambiguous
in the sense above). There MUST be questions that are unanswerble without making
additional assumptions. We say that such questions are independent of our axioms

And we know of many independent questions in mathematics: questions that simply
cannot be answered one way or the other from the axioms we use for modern math.
Some have been subsequently assumed (the Axiom of Choice). Others are still being hotly
debated (Continuum Hypothesis). Others are obscure even to most mathematicians (Martin's
axiom).

For ALL of these, we get different theories of *mathematics* by either assuming them
OR assuming them to be false. And *both* ways are equally consistent internally.

In a sense, these questions have no actual truth value, even though they are meaningful
and some are even important. You can use either the statement or its negation equally
well and have a mathematics that is equally consistent either way.

Bummer. Well, sort of.

Now, I have been pondering. We all make assumptions about the 'real world'. Among those
is that we are not the only conscious being, that there is an external world that our
senses can perceive, however imperfectly, that our memories are reliable to at least
some extent, etc.

We use these assumptions to move around in the world, learn how that world works, etc.
We develop science and technology, etc. We make art, music, and love others.

Then comes the issue of religion.

People have many different views of what things are true in their religions. There are
multitudes of different religions (especially through time) with huge variations on what
was believed, thought important, etc.

What if the existence of a supernatural is simply inconsistent?

What if the existence of a supernatural is neither provable nor disprovable from our other
assumptions? What if the question really doesn't have a truth value at all?

Now, if this is the case, what are the advantages and disadvantages of the assumptions either way?

Mathematics is actually less proven than you think. Yes, granted, like science it is intended to be proven. But then in the last century or so, we have had New Math, Common Core, and branches of math that clearly haven't been tested. Of these, Order of Operations is particularly galling. I'm gonna skip Geometry and show how math has gotten imprecise.

You see, Order of Operations is a nice theory, but seldom works in actual practice. If you are a school teacher and need to average grades, your best tool is not in fact an expensive calculator, but a cheap solar calculator. Same for library work.

Why? Because Order of Operations turns a simple average, say...

84+86+85/3

into 84+86+(85/3) or in other words, that becomes a fraction.

Suppose I'm doing a word problem, and they tell me that I have 2 apples and another grocer sells me 4 more, and by the end of the day, I sold half. How many apples are left?

Using basic logic, math should be consistent, meaning I have six apples, sold half, and now have three (3) apples.

Using elementary school math, 2+4/2 is 3.

Using parentheses CORRECTLY (2+4)/2 is 3.

Using Order of Operations, they expect you to believe that 2+4/2 is four. Any grocer who is looking at the remaining apples is like BULL****.

Effectively, Order of Operations, doesn't work for averages, and it does work for most basic math. It is the equivalent of defaulting to 2+(4/2) which nobody asked you to do.

And so, I'm going to offer a revised theory: the PEARL (named after my niece) Method of Mathematical Order.

Parentheses
Exponents
And
Right from
Left

(It should be left to right but I rephrased it to make it work as an acronym)

Basically, you solve inside the parentheses, then you square, cube, root, etc the parentheses, then you work left to right, making parentheses as you go to avoid sloppy association. Like a comma in a sentence "Let's Eat Grandma", there is no excuse to ignore making parentheses.

Using my theory, 2+rt9/2 is easily solved

Exponents first rt9 is 3
(2+3)/2 is 2.5

Unless, that last bit is a fraction, in which case, you would associate

2 + (rt9/2)

or 2 + (3/2)

or 2 + (1.5)

or 3.5

But unless noted as a fraction, it's unclear.

 

[cladking]

I think that what we're seeing here is that everything is dependent on assumptions and perspective.

There is a reality that does or does not include a Creator but we have no means of determine the answer to this nor to determine the nature of reality outside of our assumptions.

I tell children to be careful what they believe because they become their beliefs. The world they perceive and discover will be a reflection of their beliefs.

Math, science, and reason are the "sole" means to real understanding of reality but there are more than one math and one science. There is only one "logic" but it is mostly invisible to us. One must choose a perspective but it is possible to consider other perspectives through modeling. From other perspectives math might be the one reality. "Reality" is only ephemeral when assumptions change; ie-it's not reality changing but our axioms.

 

[Polymath257]

Mathematics is actually less proven than you think. Yes, granted, like science it is intended to be proven. But then in the last century or so, we have had New Math, Common Core, and branches of math that clearly haven't been tested. Of these, Order of Operations is particularly galling. I'm gonna skip Geometry and show how math has gotten imprecise.

You see, Order of Operations is a nice theory, but seldom works in actual practice. If you are a school teacher and need to average grades, your best tool is not in fact an expensive calculator, but a cheap solar calculator. Same for library work.

I would say you don't need either. You need a brain and some practice doing calculations.

Why? Because Order of Operations turns a simple average, say...

84+86+85/3

into 84+86+(85/3) or in other words, that becomes a fraction.

Which means you are using the calculator incorrectly. Part of using any machine is knowing how to use it correctly.

Suppose I'm doing a word problem, and they tell me that I have 2 apples and another grocer sells me 4 more, and by the end of the day, I sold half. How many apples are left?

Using basic logic, math should be consistent, meaning I have six apples, sold half, and now have three (3) apples.

Using elementary school math, 2+4/2 is 3.

No, it is 4, but that isn't the correct formula for that problem.

Using parentheses CORRECTLY (2+4)/2 is 3.

Yes. So? You have to use the notation correctly or you will get incorrect answers. There are times when 2+(4/2) is the answer you want.

Using Order of Operations, they expect you to believe that 2+4/2 is four. Any grocer who is looking at the remaining apples is like BULL****.

Effectively, Order of Operations, doesn't work for averages, and it does work for most basic math. It is the equivalent of defaulting to 2+(4/2) which nobody asked you to do.

You are assuming that averages is the only reason this would come up. The point is that you want to add all the numbers *before* you divide, which means you want
(2+4)/2.

And so, I'm going to offer a revised theory: the PEARL (named after my niece) Method of Mathematical Order.

Parentheses
Exponents
And
Right from
Left

(It should be left to right but I rephrased it to make it work as an acronym)

Basically, you solve inside the parentheses, then you square, cube, root, etc the parentheses, then you work left to right, making parentheses as you go to avoid sloppy association. Like a comma in a sentence "Let's Eat Grandma", there is no excuse to ignore making parentheses.

Using my theory, 2+sqr9/2 is easily solved

Exponents first sqr9 is 3
(2+3)/2 is 2.5

And that would be one possible alternative, but it tends not to be a good one when going further than simple arithmetic. For example, a simple expression like

x^2 +3x +2

would be interpreted as
(x^2 +3)x +2

which is a completely different thing. Or, we could write it as

x+3x+3

which would be interpreted in your scheme the same way as x^2 +3x+2 is in the usual one,
but would hide the fact that it is a quadratic equation and thereby hide the best ways of dealing with it.

So, while your scheme is at least unambiguous, it fails to be useful at even the next level. The standard scheme was designed, not for taking averages only, but for much more complicated situations where the standard scheme works immeasurably better. And it was arrived at through trial and error finding what works best.

 

[Polymath257]

What you are describing is related to the agnostic point of view: that nothing can be known, claiming neither faith nor disbelief.

Not quite. That would be simply the recognition that some position is independent: that it cannot be known.

Faith would be the further step of adopting one alternative to add to the list of axioms.

 

[Samantha Rinne]

This is liberalism, and it's infiltrating math and science. It's basically the equivalent of cultural relativism.

"This culture or religion practices murder, theft, and slavery. We should just accept that their culture is different." No, we need to accept they are a culture of murderers, thieves, and slavers.

What ought to be taught is that you have consistent assumptions given correct notation. However, if you were to explain to a teacher that Order of Operations doesn't actually work, and that math ought to clearly show a difference between division and fractions, you'd get a C or worse for your efforts.

What is required is a higher standard for teachers, to actually make sure that they know the how and why of math, not just blindly accept proofs that are unproven.

When a teacher gives their students a math problem 2+rt9*sq7/5 they have failed them already. They are expecting them, without any advice on how to associate, to solve a complex problem where the order is assumed but not known. Is it 2+((rt9*sq7)/5) or 2+ (the fraction of (3*49)/5)? Is it (2+3)*(49/5)? Or is it ((2+3)*49)/5?

A teacher must show clearly to their students what is right and what is wrong by narrowing any possible assumptions.

Polymath, yes there are times when that is what you want.

But at all times, you should avoid sloppy math. Just as I don't write a sentence and go on and on without any commas without any breaks of any sort while I list almonds raspberries and pears just as I use commas to separate paired items like peanut butter and jelly from the rest of a list which is why the last of a list should always be comma and just as this sentence sux without any commas math with no parentheses makes it difficult or impossible to reliably perform the problem. This is what a math problem like that would look like as a sentence. It's ****. And Order of Operations uses a procedure that is not common to either basic elementary school math or actual experience using real objects, (in the apple example, the grocer would know something is funny about that math) to get answers that don't apply.

It's not about using the calculator correctly. The solar calculator comes to the correct assumption because it DOESN'T USE Order of Operations. The graphing calculator does, and fails a simple math problem. And no, the correct answer is NOT four. The grocer would have gotten 6 apples by the end of his buy period, and sold exactly half of those six. He would be able to look at those three apples left, and without any math besides counting, tell you there are three apples.

In the grocer example, yes if they halved the ones bought from the other guy, then it would be 2+(4/2). But this is why we HAVE parentheses. So we can use them to tell how the math problem should go. When we don't, it should ignore everything and proceed left to right.