h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

I think that I'm inclined to agree with Pc here, but to add that since maths can represent the territory as consistently and successfully as it does, it would seem that there must be some real feature(s) of the territory that are being represented.

Noggin

<...it would seem that there must be some real feature(s) of the territory that are being represented.>



Pattern-chaser

"Who cares, wins"

PC: <>

<< <...it would seem that there must be some real feature(s) of the territory that are being represented.>

>>

At a guess I would say that you're getting caught up between the language and the thing the language represents. As I've said before, the symbols of math are all arbitrary and created, used to express something in the real world. The symbols 2+2=4 have no existence other than as shapes on your screen. But the symbols represent something very real, an idea you appear to support. By the same token, the orange exists, even though the symbols o,r,a,n,g, and e, and the vocal sounds we associate with them, are otherwise meaningless.

Math as a set of symbols, then, is all map. Math as a collection of operations and functions which produce universal results, regardless of the symbols used, is all territory.

It seems that people are saying that mathematics itself is a territory, which can be explored and mapped out, which is what the mathematicians are up to. It's a different kind of territory from Greenland, though, being accessible mentally rather than physically. The fact that different explorers, each accessing it via their own mind, keep reporting the same findings - that's why we know there's really a territory there, right?

Meanwhile, knowledge of that mental territory, as informed by our maps of it, contributes to the making of maps for other, physical territories, which sometimes end up working really well. This success makes it seem likely that the territory has features that really are mathematical, although that can't be proven just from our maps. Or does that constitute a proof? Or did that question make sense?


GTB

"Ceci n'est pas une pipe" The label/thing and map/territory concepts are intimately associated, if not exactly the same thing. They can prove difficult to disentangle.



Once again I ask: If mathematics exists in the real world, then show it to me.

You can describe your emperor's clothes until you're blue in the face, but if you can't *show* me those clothes, perhaps there *are* none, and your emperor is buck naked!

Show me mathematics in the real world.

Pattern-chaser

"Who cares, wins"

The territory - in this special case where the territory is the real world - is ultimately unknowable. Certain mathematical processes resemble the territory so closely that ... we can use them as a map to help us navigate it! Whether the maths and the real world are similar or congruent, we cannot tell. It doesn't matter. A good map proves itself by facilitating successful navigation of its subject.

Pattern-chaser

"Who cares, wins"

Here's a thought:

Maths is abstract, but it can be manipulated in our brains. This manipulation requires some form of representation of the relationship between mathematical ideas.

So...is Maths neurochemistry?

And the neurochemical patterns that arise from maths can also be represented on paper, which provides a mechanism for transferring them between brains.

Incidentally, I once had a friend (sadly now dead) who had a brain tumour removed and lost her ability to count backwards. Her comment was 'There goes my career as an astronaut.'

GTB: That's pretty much my position. I'd argue, however, that it is accessible physically. It just can't be expressed in any other language except the language of math. A lot of math comes about from physical observations, and then we find ways to express them. For instance, you can measure a lot of triangles to come up with Pythagorean Theorem.

PC: <>

Surely you use mathematics on a daily basis. You use the symbols and come up with results that have value to you in the real world. Every time you use the language of mathematics, you're using its cartographical function, and by using it you're navigating successfully through its reality. This insistence on physical evidence of mathematics is like someone navigating the coast of Greenland, finding the physical features exactly where the map says they should be, and then declaring, "There is no Greenland!"

Guys this has gone just a bit too far, I mean, really:

"Maths is abstract...", and "So...is Maths neurochemistry?"

I am not a number, I am a free man!

Blessings,
Matholwch /|\

Nonsense, Math. (sic!)

If someone throws you a ball, and you intercept its parabola to catch it, then your brain will be doing maths *somewhere* within its jellylike mass. It will have neural processes which allow the path of the ball, and the intercept path of your hand to be represented. That's maths.

The (let's call it) 'higher' maths that PC et al have been talking about is merely represented in different parts of the brain. That's why it has to be learnt in school, rather than by playing catch with Daddy in the back garden.

Let's take another example: you can multiply, right? But you have no innate mechanism for doing so. You (probably) *do* have innate mechanisms for simple addition and subtraction. At school, you learnt the concept of number and extended this to be able to add and subtract all the base 10 digits...and this is largely a memory process. And then they extended this by encoding within your neural pathways a look-up table for multiplication that we call 'the times tables' (And, doubtless in your day they later took it further by teaching you to use log tables, etc. etc. whereby your lookup processes were supplemented by print.)

What was happening here is that methematical concepts were being burnt into your memory. Professional mathematicians simply have more concepts. BUT - we all have some.

Me...as you know, I'm merely a collection of some base animal processes. Nowt wrong with that.


btw...
Following the Great Flood, Noah was a releasing all the animals, two by two. He took two snakes and put them on the table and said:
'Go forth and multiply!'
But the snakes said:
'We can't! We're adders!'
So Noah said:
'That's OK...It's a *log* table!'

Wow.

I'll be telling that one to everyone in the math department who'll listen next week. Any complaints, I'll send your way, Ed.



Is maths neurochemistry, is maths neurochemistry? Tricky. A familiar name in these halls, Lakoff - he wrote a book saying yes, it is. I didn't get the impression he knows all that much mathematics.

Is the independence of the parallel postulate from the other axioms of geometry a fact for all humans, or is it a fact for all possible reasoning minds in the universe? (Is it even a fact for any more than the few people who know what it means? Or the even fewer who can prove it?)

If mathematics is universal, how could we know that for sure? Maybe its great success in physics is evidence...


just thinking aloud; no answers here...

GTB

<>

Neurophysiology might be a better term than neurochemistry, given that the physical structure of the brain is also going to be a factor.

I don't think detailed knowledge of mthematics is actually necessary to make (or deny) a claim that the patterns of mathematics are embodied in the brain, because one could make or deny such a claim about other aspects of human knowledge too.

Noggin

I know this is a circular argument, but after being honed by 500 million years of evolution, shouldn't our brains have programs which reflect 'reality' pretty accurately? After all, a dog catching a frisbee must somehow be able to calculate where the frisbee is going to be, so it's not like human-type intelligence is a huge factor here.

Mathematics contributes to excellent and useful maps, and yes, I do use it every day. But I don't navigate "its reality", I navigate the real and natural world.

My insistence on (the total lack of) physical evidence of mathematics in the real world is intended to illustrate that maths is part of the map not the territory.

What is this about Greenland? You navigate its coast, quite successfuly, using a map. That's what maps are for. The map exists and so does Greenland.

But look more closely at the map. It has contour lines. [It has other symbols and features too, but let contour lines stand as an example of all of them.] Now look from your boat toward the shore. That's Greenland. There are no contour lines drawn on it, are there? And yet you can use contour lines to make a fairly accurate estimate of the height of that headland, can't you? Contour lines are a purely map-based phenomenon. They don't exist in the real and natural world. But they're useful and they work. Just like mathematics.

When I ask you for evidence of mathematics in the real world, I'm saying to you:

Look, there is no mathematics in the real world. If there was, you could provide me with evidence, couldn't you? I offer this lack of evidence as evidence () that I am correct in this, and you are mistaken.

- Over there I see 'two cows' but I can't see 'two'. Where is 'two'? In our heads.

- I drop a pebble from a height of five metres and it takes just over a second to hit the ground. But where are 'five', metres' and 'a second'? In our heads.

- The pebble is there, and we both watched it fall and hit the ground, but where are the units and measurements? In our heads.

They're map-based. They do exist, and they're clearly useful, but they don't form part of the real world. They exist in our heads. This is the difference between the map and the territory, in our *very specific* example where the map is our mental model of the real world, and the territory is the real world itself.

Pattern-chaser

"Who cares, wins"

<> P-C

What is mathematics anyway? If it's x + y =z, then of course there isn't. But if it is defined as relationships or ratios, then it wouldn't be visible, would it? And would that make it any less real?

Hi P7!

My point is not that maths is non-existent because it's not visible! My point is that there is no maths in the real and natural world. [There are aspects of the real-world whose functioning closely resembles certain mathematical processes. But resemblance is not existence.] Maths DOES exist in our heads, though, and this is about as real as anything gets!

I am trying only to distinguish between the map and the territory in the specific example where the map is our mental world-model and the territory is the real world. The difference between the two seems important to me.

[When this unexpected deviation into whether or not maths is map-based is over, the next interesting point would seem to be: *why* does the difference between the map and the territory matter?]

Pattern-chaser

"Who cares, wins"

Does anyone mind if I quietly bang my head against the wall? You guys can carry on...















*bang*

<>

But if the rules used to generate the map are the rules used to generate the territory....then *does* the

As far as I know, the territory is the master, the reference by which maps and the like are judged. And (also as far as I know), the territory is not generated by rules, it just *is*. The rules are in our heads. ... Like maps.

Pattern-chaser

"Who cares, wins"

*bang*