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

I'm not sure that the map-territory thing is an analogy, I think it's more or less literal, with both terms being used in a general sense. The world-model in your mind is a map of sorts, and the territory that map describes is nothing more or less than the real world. I think that maths is map-based, not territory-based. There is no maths in the real world, but there's plenty of it in some very successful maps.

Pattern-chaser

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I think that it's complicated because maths is at one and the same time a co-ordinate system for mapmaking, and at the same time a part of the map itself (that is we have a map of the territory "mathematics").

As for whether it is invented or discovered, I think it's a bit of both. Mathematics is a branch of metaphysics, an important part of which concerns itself with the organisation of the human mind - in a sense we are discovering something in the mind, rather than something in the world.

I would also add that this structure of the mind has not been honed just by a few thousand years of *conscious* trial and error, but (if our current map of this particular territory is a good one), by millions of years of *unconscious* trial and error in the form of evolution and naturl selection.

Noggin

The map-territory relationship still holds, of course, but there is a qualitative difference between our example, where the territory is the real world, and the 'territory "mathematics"' where the territory is definitely not the real world, or even part of it.

Pattern-chaser

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<> P-C

If maths is about relationships or patterns, then even though there are no equations writ large (literally) in nature, doesn't mathematics have *some* kind of existence? As an analogy, if the pattern of activity in my brain right now leads to the thoughts I'm thinking, what kind of existence do my thoughts have? They are not material in one sense, being at one level just the movements of various ions and proteins through my brain, but it is more useful and informative to describe the output of those processes rather than what actually underlies them.

Perhaps it would be easier to eschew mathematics as a whole and concentrate on a subset - I suggest "Euclidean geometry".

Now there's no doubt that most, if not all, humans (except some brain damaged individuals) perceive themselves as existing in a "Euclidean Space" even if they don't know that that's what it's called. This space is produced unconsciously, and is a co-ordinate system for the map of our physical surroundings. But it is also possible to make a map of the co-ordinate system itself. Such a map is not directly a map of the "territory", but it is a map of the possibilities of the territory. It's also a map, in some way, of some rather fundamental orgnisational features of the mind.

Noggin

My poor brain is trying to escape through my ears

*wanders off mumbling to herself*

P7:

Oh yes! It exists in the minds of humans, but not in the real world. [Unless you want to split hairs, and observe that humans exist in the real world, and so their minds, and the contents of those minds, must also exist in the real world. Personally, I think this is a defensible argument, but hugely non-useful, and should not be pursued.]

Pattern-chaser

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Ok, this may be a really stupid question, but say there are 2 numbers, say 2,938,833,344 and 83,920,202 and no one's ever thought to sit down and add them together, doesn't the answer that you would get exist whether someone has ever worked it out or not?

Hi Noggin

What you say is all true, as far as I can tell. However, some of the points I have been trying to make refer to a *particular* map-territory pair: the map ('world-model') inside the mind of a typical human, and the territory (the 'real world') to which it applies.

In my example, it is correct to observe that mathematics is map-based, and has no presence in the corresponding territory. It is *not* correct when applied to territories that are part of people's world-models, even though the map-territory relationship still applies!

Pattern-chaser

"Who cares, wins"

<>

But do numbers in this abstract sense exist at all? I'd say that the answer exists/doesn't exist in the same sense that the original numbers (and the rules of addition) exist/don't exist. On the other hand if you had 500 objects and 300 objects, there would be 800 objects whether anybodycounts them or not.

Noggin

Agreed, Nog. You'll have 800 objects whether you count them or not. Those 800 objects reflect the territory (the real world), rather than the map.

To get there, first we have to define our terms. What is an object? What is 800? At this stage we're in the realm of the subjective, because we're talking about creating a language to describe what we're seeing. We see the territory, and we're mapping it.

Once you've gone through that process, you evolve a little further. Perhaps you have some purpose for those 800 objects. Let's say the objects are oranges, and your purpose is to feed people. How many people can you feed with 800 oranges?

Now, somewhere along the line you've defined "oranges" as being a round, skinned thingy that falls from a particular type of tree. You defined "people" as more than one member of homo sapiens, and you've defined "800" as the shorthand symbology for some quantity using at a minimum a base-9 numbering system (assuming 1-7 are valid symbols). We're still in mapping stage here.

So now you figure out how many oranges each person will consume and divide 800 by that figure. No matter what numbering system you use, no matter what definitions you used to define "orange" and "people," the numbers come out the same. The base-16 answer is the same as the base-12 answer, because even though the symbols look different, the same number of people will still be fed.

The answer never changes no matter what map we use. Maps are under our control, and we can change them whenever we need to. We can't change the territory. By this definition, the math used to calculate the answer is a part of the territory.

A friend of mine who is a theoretical physicist told me an interesting fact about quantum field theory. It may be that two different observers would experience a different number of particles existing in the same place. Suppose A is accelerating extremely fast relative to B, but that A and B are in the same place at some particular time, then A will see many more particles of some particular sort than B will. So, it may not be true to say that if there are 500 objects and 300 objects then there are 800 objects whether anyone counts them or not. Who is doing the counting is crucial.

My take on this, using P-c's specially developed vocabulary, is that maths is in itself neither map nor territory, it's more like what Blathers said - it's one of the tools you use in making the map. The direct applicability of arithmetic to 'ordinary sized objects' (like oranges, trees, etc.) is a map, but arithmetic as a branch of maths exists even without being applied to counting ordinary things. As the examples of the funny number of particles that different observers see shows, arithmetic doesn't apply universally. In the strange quantum and relativistic worlds that modern physics hypothesises, this sort of thing shouldn't surprise us at all.

Of course not; the "answer" comes from the real world; the reference; the territory. So of course it doesn't change if the map does.











Whoa! Wait a minute! What definition is that? That it's a map if it's under your control, and you can change it, otherwise it must be part of the territory? I don't really think that works, do you?

Pattern-chaser

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P-C said:

>>

>>

Isn't that the problem with the religious? The curious notion that you can define the territory by choosing your own brand of map?

If you Google for "map territory" I think you'll find that this vocabulary is far from just mine. I have been referring to a specific example, though, where the map is the world-model in someone's head, and the corresponding territory is the real world.

For simplicity, if something is used in the creation of a map, might it be easier to label it as part of the map, or maybe use a phrase like 'map-based' or something? The essential distinction is between map and territory, so association with one or the other establishes the basic relationship.

If maths was part of the territory - the real world - then we would be able to see it, wouldn't we? All other arguments aside, if maths is part of the real world then show me an element - any element - of maths in the real world. That ought to be possible. Any takers?

Pattern-chaser

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Pattern-chaser

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Hi Ed!



You can change what the territory looks like (to you) by selecting an appropriate map, just as a physical map shows different things to a political map. The territory doesn't change though.

Pattern-chaser

"Who cares, wins"

Well, yes, and the only sensible thing to do is to be prepared to throw the map away.


There was a neat exaqmple a while back when R4's flagship consumer moaning programme, Ewan Jaws, uncovered the outrageous scandal that Ryanair's advertised flights to Copenhagen didn't even land in Denmark but in Sweden. Michael O'Leary pointed out in his usual testy way that there's a new bridge and the airport is only 30 minutes by bus from Copenhagen. A case of using an out-of-date map which caused the moaners to look at things politically rather than physically.

There is no One True Map, only maps that facilitate navigation and maps that don't. The latter are the ones we throw away.

Pattern-chaser

"Who cares, wins"

PC: <>

To be more specific, you can't change the territory simply by changing your mind. You can change the orange by peeling it, juicing it, or detonating an explosive charge inside it, although even then it's just an orange, in a different condition. But you can't just decide it's a peach.

Ed: <>

Well said.

PC: <>

The problem is there are some maps circulating which lead directly to conflict and strife, spreading human misery wherever they go. But these maps promise the intrepid explorer the equivalent of El Dorado or The Fountain of Youth. And despite the many disastrous explorations, the maps are still considered too precious by the idealistic and the foolhardy.

The most common maps in Western society lead nowhere, but people continue to navigate by them regardless.

PC: <>

BtM:

I'd worked *that* out for myself. My problem is when you turned several reasonable and accurate observations into a "definition", and drew a conclusion that seemed unjustified to me:

BtM:

Your thesis seems to be:

We can't change the territory.
The maths gives an answer that doesn't change.
Therefore the maths is (part of) the territory.



If maths is part of the real world, then please point it out to me. If it really and truly exists there, then this shouldn't be a problem, should it?

Pattern-chaser

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