Is mathematics invented or discovered?

[nomadreid]

One of the possible false assumptions here is that mathematics is uniform enough to have the same answer for all of it. It is possible that parts of mathematics are invented, and parts are discovered. The people claiming one side often latch on to that part of mathematics that fits their choice to justify their claim. In this way the debate could go on uselessly forever.

 

[PeroK]

I'm not sure I agree that group theory was invented. The axioms of group theory were chosen the way they are because it leads to relationships between transformations that describe objects in our universe. In this sense I would argue that Galois saw that those axioms led to particularly useful and interesting structures. If it were purely invented, then you could choose any random set of axioms for your theory.

The monster group isn't an object in our universe.

 

[DaveC426913]

In this sense I would argue that Galois saw that those axioms led to particularly useful and interesting structures. If it were purely invented, then you could choose any random set of axioms for your theory.

You just self-corrected.

You could "choose any random axioms for your theory", but you don't - and the reason you don't is because certain "axioms lead to particuarly useful and interesting structures".


Look, all types of categorizing are human-invented. We might group green things together: leaves, caterpillars, bird plumage. But that is not a natural grouping. We invented that group by deciding that what was important was colour*. There is no such thing as a green group in nature. The same thing can be said about any form of grouping. The group's definition is defined by what some human decides is significant.

*actually, even that doesn't exist objectively. "Green" only exists in the mind.

 

[phyzguy]

The monster group isn't an object in our universe.

I agree it is not a physical object, but it is a well defined structure. Where does that structure reside if it is not in our universe?

 

[phyzguy]

Look, all types of categorizing are human-invented. We might group green things together: leaves, caterpillars, bird plumage. But that is not a natural grouping. We invented that group by deciding that what was important was colour*. There is no such thing as a green group in nature. The same thing can be said about any form of grouping. The group's definition is defined by what some human decides is significant.

But that's exactly my point. "Some human decides what is significant" by studying the objects around us. As you say, we are guided by "what is a natural gouping". We don't just make that up (i.e. invent it), we are guided by inventorying the things around us (i.e we are discovering them).

The structure of group theory was guided by how things transform in our universe. It wasn't just made up on a blank slate.

 

[BillTre]

natural grouping

How are you defining a natural group?

 

[martinbn]

I'm not sure I agree that group theory was invented. The axioms of group theory were chosen the way they are because it leads to relationships between transformations that describe objects in our universe. In this sense I would argue that Galois saw that those axioms led to particularly useful and interesting structures. If it were purely invented, then you could choose any random set of axioms for your theory.

I don't think this is historically accurate. He did not give the modern definition of group. For him groups were groups of permutations.

 

[martinbn]

My question is: what diffetence does it make?! No matter what the answer is, it will not change how we learn, teach or do maths, will it?

 

[DaveC426913]

How are you defining a natural group?

Something that exists distinct from humans.

A pack of hadrosaurs might be considered a natural group. Humans did not invent the peculiar bond between herd members. There are physical consequences to that particular proximal collection of things.

The prime numbers are not a natural group. Just because there are things in nature that have seven elements and others that have eleven elements, it does not follow that they are a group. We have labeled them as primes because we see them as sharing mathematical properties (they are only divisible by one and themselves), but that does not make them part of a natural group.

 

[DaveC426913]

My question is: what diffetence does it make?! No matter what the answer is, it will not change how we learn, teach or do maths, will it?

Ergo: this is a philosophical question and is thus in GD.

 

[PeterDonis]

We have labeled them as primes because we see them as sharing mathematical properties (they are only divisible by one and themselves), but that does not make them part of a natural group.

What about the ways prime numbers show up in "natural" settings?

For example, there are 13-year and 17-year cicadas. It is not a coincidence that both of those numbers are prime: there is evolutionary logic behind it. But humans had nothing to do with how those cicadas evolved or why the times involving them are what they are.

I don't think the distinction you are trying to make is very clean.

 

[DaveC426913]

What about the ways prime numbers show up in "natural" settings?

For example, there are 13-year and 17-year cicadas. It is not a coincidence that both of those numbers are prime: there is evolutionary logic behind it. But humans had nothing to do with how those cicadas evolved.

Agree. Those are prime, and we see how it got that way.

My point here is that nature does not have a "group" of primes that it picks from. The group of primes comes from a mind that has insight into why these two numbers can be considered to have similar properties.

BTW, "evolutionary logic" is an anthropomorphic pittrap, as you're aware.

I don't think the distinction you are trying to make is very clean.

It is not clean, but I don't think it follows that math is discovered. That's non sequitur.

 

[PeterDonis]

nature does not have a group of primes

Primes aren't a group, mathematically speaking. They aren't closed under either addition or multiplication, the two operations present in the underlying ring.

 

[PeterDonis]

"evolutionary logic" is an anthropomorphic pittrap

I disagree. We conceptualize it in a particular way, but evolution does what it does whether humans conceptualize it or not, and what it does is not random, it has patterns in it, whether humans are there to recognize that fact or not. Evolution was working for billions of years before there were any humans to conceptualize anything. Don't confuse the map with the territory.

 

[PeterDonis]

I don't think it follows that math is discovered.

I agree; I think the distinction between "discovered" and "invented" that this thread is predicated on isn't clean either.

 

[DaveC426913]

Primes aren't a group, mathematically speaking. They aren't closed under either addition or multiplication, the two operations present in the underlying ring.

I'm not using the term "group" in the direct mathematical sense (becaue I think it would get too murky). I'm simply looking at any type of abstraction as the product of a thinking mind.

A hadrosaur herd is objectively significant without any existence of a human mind.

 

[DaveC426913]

I agree; I think the distinction between "discovered" and "invented" that this thread is predicated on isn't clean either.

My distinction is:
Discovered: it objectively existed as part of the world before humans came along.

Does anyone want to hone that?

 

[PeterDonis]

Discovered: it objectively existed as part of the world before humans came along.

Then at least some prime numbers are discovered--those that appear, for example, in the time periods of cicadas.

 

[PeterDonis]

I'm not using the term "group" in the direct mathematical sense

I don't think your usage is consistent with others in this thread--they appear to me to be using "group" in the direct mathematical sense, for example in talking about the monster group and Galois.

I would suggest using a different word for what you mean by "group" to avoid confusion.

(becaue I think it would get too murky).

Huh? The mathematical concept of a group is perfectly well-defined. I'm not so sure about your concept of "group".

 

[DaveC426913]

Then at least some prime numbers are discovered--those that appear, for example, in the time periods of cicadas.

I'm not talking about the numbers themselves; I'm talking about "The" prime numbers - i.e. that 13 and 17 are part of a group. The group - "the primes" - was invented.

 

[BillTre]

A hadrosaur herd is objectively significant without any existence of a human mind.

These have only been observed indirectly (fossil footprints).
It would seem to be more concrete to discuss things that are more directly observable, like herds we can see today. I am more familiar with herds of fish ((schools) so I am switching to that now. Fish schools have been studied a lot. They are not all thee same. Some schools are very tight, tightly packed and turning in unison. Others are very loosely grouped and based on temporary relationships. The fish come and go from the group.

Other natural groups have been based on biological phylogenies, clades of taxa (taxonomic groups, usually species). This makes sense in that they are all derived from a common ancestor and grouped in that way (phyllogenies has also been displayed as nested Venn diagrams). However, biological complexity blurs even that since reticulate phylogenies (both branching apart and together) and known. A possible way around it would be to include all living things (presumably related to the Last Common Universal Ancestor (LUCA), however even that could be challenged by possible reticulate phylogenies prior to LUCA. .

Another commonly cited natural group in evolutionary biology is species vs. higher taxonomic groups, which are considered human derived divisions at larger phylogenetic scales (genera, classes, families, phyla, etc.). This runs into problems with: defining what a species is and were do isolated breeding populations fit into it. Some consider breeding populations are a more basic (and natural?) group than species (which ca become seperate and then evolve independently.

Would a cell be a natural group of its component parts, even though many parts are only transiently a part of the cell?

 

[DaveC426913]

I don't think your usage is consistent with others in this thread--they appear to me to be using "group" in the direct mathematical sense, for example in talking about the monster group and Galois.

Agree. But I'm not sure it invalidates my point.

Can someone show me the monster group in-the-wild? Not component of it; bnot elents that you surmise belnig in it. Can you show me the group itself? Would Tarzan be able to look at it (i.e. a human without a grounding in math)?

If not, it's abstract. It was created subsequent to the creation of the human mind.

I would suggest using a different word for what you mean by "group" to avoid confusion.

Never mind. I'll drop that argument.

 

[DaveC426913]

These have only been observed indirectly (fossil footprints).

My point exactly. So objectively not part of any human abstraction.

Individuals in the herd behave in particular way because they are a group - strangers are not part of the group. It's a natural, concrete, distinction - between these things and those things - and it requires zero abstraction.

It would seem to be more concrete to discuss things that are more directly observable, like herds we can see today.

I picked hadrosaur on purpose to show that it objectively existed long before humans decided that herds were an interesting property of a group of animals.

I am more familiar with herds of fish ((schools) so I am switching to that now. Fish schools have been studied a lot. They are not all thee same. Some schools are very tight, tightly packed and turning in unison. Others are very loosely grouped and based on temporary relationships. The fish come and go from the group.

Sure.

Another commonly cited natural group in evolutionary biology is species vs. higher taxonomic groups, which are considered human derived divisions at larger phylogenetic scales (genera, classes, families, phyla, etc.).

Yep.

 

[BillTre]

I picked hadrsoasaur on purpose to show that it objectively existed long before humans decided that herds were an interesting property of a group of animals.

Seems like it is completely a human made up thing.
It is not objective that this group existed as a group based on some footprints, likely maybe not a solid conclusion.

 

[DaveC426913]

I disagree. We conceptualize it in a particular way, but evolution does what it does whether humans conceptualize it or not, and what it does is not random, it has patterns in it, whether humans are there to recognize that fact or not. Evolution was working for billions of years before there were any humans to conceptualize anything. Don't confuse the map with the territory.

I am on no way suggesting it is random. Please don't put that on me. I'm not some Creationist. Nothing I said should warrant that response.

 

[DaveC426913]

Seems like it is completely a human made up thing.
It is not objective that this group existed as a group based on some footprints, likely maybe not a solid conclusion.

Eh. Fine.
I thought we could distinguish between the confidence in paleobiology and the confidence in pack mentaility but OK, let's use an extant species.

Now that we're using an extant species, the onus is you to to how that its group label is not a result of humans labeling it to be so. That's what I was trying to avoid, but sure.

 

[BillTre]

the onus is you (as) to how that its group label is not a result of humans labeling it to be so. That's what I was trying to avoid, but sure.

No it is not.
Its up to you to show how your claim applies to the more realistically observable cases you just agreed to. Making claims about an unobservable case but avoiding making claims about a more concrete case...?

 

[PeroK]

I agree it is not a physical object, but it is a well defined structure. Where does that structure reside if it is not in our universe?

It could be an invention of the human mind, via the formalisation of abstract mathematics.

 

[DaveC426913]

No it is not.
Its up to you to show how your claim applies to the more realistically observable cases you just agreed to. Making claims about an unobservable case but avoiding making claims about a more concrete case...?

All right.

The wolves act like a pack in tangible ways: inter-pack dynamics, pecking orders, etc. This is distinct from how they act around non-pack wolves, and they have been doing so since before we came along to apply groups to things. There is nothing abstract about the group that is a wolf pack. Even the wolves know they are a group.

Wolf packs - groups of wolves - objectively existed as part of the world before humans came along. i.e discovered.

I do not think the monster group itself can be conceived of, let alone seen, independently of a human mind. Especially since - unless I'm misunderstanding - it is a group comprised of abstract mathematical objects.

 

[phyzguy]

It could be an invention of the human mind, via the formalisation of abstract mathematics.

So from that standpoint, when did this invention of the human mind come into existence?
(1) When Galois first wrote about group theory?
(2) When people started studying simple groups?
(3) When someone first realized that this huge group existed?
(4) Since obviously nobody has conceptualized the monster group as a whole, does it exist even now?