After submitting my last post, I was thinking to myself, what example would I give if someone now asked what new mathematics had been developed in response to needs from physics. Calculus was going to be my example. Calculus is the mathematics describing change, when that change occurs in a reliably predictable way. Change, of course, has been observed ever since there have been observers.

(EDIT: Also, of course, that change has to occur in a smooth way, not an abrupt, jumpy way, for calculus to work. Just in case a mathematician reads this and says, hang on a minute...)

We don't see numbers and mathematical formulae when we observe our world, we first have to have an idea of what's happening around us and then express that idea using the language of maths/physics. Maths is created to mirror our idea, but we no more discover maths than we discover ideas (about the world) in our minds. Ideas come through creative thinking and even logic is a process of creative thinking, hence maths too is an expression/language of creative thought.

Nothing in the universe is discoverable unless recognised by the mind and the mind only discovers something if it understands that thing, so discoveries in the world actually amount to thoughts in the mind. In effect, what we are observing in the world is a connected series of thoughts about it. The mind's own representation/map of it.

From the onset, minds have invented concepts and ideas to identify and understand. I say 'invented' because concepts and ideas are not observable through telescopes, they emanate from ourselves. In that sense, our minds are the source of all discovery.

jamest wrote:We don't see numbers and mathematical formulae when we observe our world, we first have to have an idea of what's happening around us and then express that idea using the language of maths/physics. Maths is created to mirror our idea, but we no more discover maths than we discover ideas (about the world) in our minds. Ideas come through creative thinking and even logic is a process of creative thinking, hence maths too is an expression/language of creative thought.

So the order of events is:

i) stuff happens in the world
ii) we observe it
iii) we formulate a description of it

This doesn't amount to 'invent' then, meaning 'to create or design something that did not exist'. The description is predicated upon the existing relationships.

As an example, you move to a new house and have a walk around the neighbourhood and find your nearest news-agent. You don't say you 'invented' the path to the news-agent, whereas you could say you 'discovered' it.

jamest wrote:Nothing in the universe is discoverable unless recognised by the mind and the mind only discovers something if it understands that thing, so discoveries in the world actually amount to thoughts in the mind. In effect, what we are observing in the world is a connected series of thoughts about it. The mind's own representation/map of it.

It's recursively redundant. The property of 'discover' is only plausibly available to a mind, but that doesn't mean the mind invents the thing it discovers. The mind then creates a map, of varying fidelity, of that external thing - again, the notion of 'invent' here seems entirely superfluous when the map is of a pre-existing quantity. We invented the paper, the ink, and the symbols to represent the sounds of words we use to label the stuff on that map, but we didn't invent the subject of the map: if that were the case, the entire concept of 'map' would be redundant.

jamest wrote:From the onset, minds have invented concepts and ideas to identify and understand. I say 'invented' because concepts and ideas are not observable through telescopes, they emanate from ourselves. In that sense, our minds are the source of all discovery.

There are indeed many things which are invented, and yes absent the central processing state of our brains, we wouldn't be able to 'discover' or 'invent' or any other concept that is predicated upon the behavior of the mind: that is clearly banal. However, when we're talking about, for example, a scientific understanding we don't claim to have invented the thing we describe.

Eureka (I find), not epinoisa (I invent). We could say, for example, that Darwin discovered natural selection - but not that he invented natural selection, despite being (among) the first to alight on the idea and accurately describe the phenomenon.

I think this contribution illustrates why I wondered, in my first post in this thread, whether very much turns on whether mathematics is discovered or invented; to put it more brutally, "who cares?"

It turns out that it depends on how you define "discover" and "invent". I come down on the side of "discover" for the reason that I stated: all the mathematics that a mathematician develops follows inevitably from the axioms which which he begins. You come down on the side of "invent", because we can't see mathematics:

We don't see numbers and mathematical formulae when we observe our world

We're not disagreeing about anything, just using words differently.

(Though this statement might be questioned:

the mind only discovers something if it understands that thing

But I don't think this statement has any bearing on the asnwer to the question in the thread title.)

EDIT: This was in response to jamest's post, not to that of Spearthrower, which arrived while I was ruminating.

the mind only discovers something if it understands that thing

You still bump into walls in the dark.

Spearthrower wrote:
Eureka (I find), not epinoisa (I invent).

Eureka = I have found (it).

Spearthrower wrote:
We could say, for example, that Darwin discovered natural selection - but not that he invented natural selection, despite being (among) the first to alight on the idea and accurately describe the phenomenon.

Indeed, and how very stupid not to have thought of it.

Evolving wrote:

Spearthrower wrote:
Eureka (I find), not epinoisa (I invent).

Eureka = I have found (it).

I wasn't trying to render it grammatically accurately, just point out that the way we think about alighting on a concept is the notion of find rather than invent or create.

Evolving wrote:

Spearthrower wrote:
We could say, for example, that Darwin [i]discovered natural selection - but not that he invented natural selection, despite being (among) the first to alight on the idea and accurately describe the phenomenon.

Indeed, and how very stupid not to have thought of it.

Did you ever see any of the history of authors predating Darwin who clearly understood the concept, but never bothered themselves with attempting to formalize it? A Scottish guy writing a book about growing the best kind of wood for boats, for example.

Evolving wrote:I think this contribution illustrates why I wondered, in my first post in this thread, whether very much turns on whether mathematics is discovered or invented; to put it more brutally, "who cares?"

It turns out that it depends on how you define "discover" and "invent". I come down on the side of "discover" for the reason that I stated: all the mathematics that a mathematician develops follows inevitably from the axioms which which he begins. You come down on the side of "invent", because we can't see mathematics:

We don't see numbers and mathematical formulae when we observe our world

We're not disagreeing about anything, just using words differently.

Agreed: whether it's philosophizing or filosofeezing about math, I am not sure I'm the best person to be doing it. To me, discover and invent are just words - they don't constrain reality but only attempt to formalize some element of it such that we can communicate a meaning, so if after using them the meaning is still poorly communicated, it's the fault of the words not indicative of some deep truth about the universe.

Both have some bearing, but both seem to mean something they shouldn't properly mean with respect to how math.

Spearthrower wrote:

Evolving wrote:

Spearthrower wrote:
Eureka (I find), not epinoisa (I invent).

Eureka = I have found (it).

I wasn't trying to render it grammatically accurately, just point out that the way we think about alighting on a concept is the notion of find rather than invent or create.

I know.

Spearthrower wrote:

Evolving wrote:I think this contribution illustrates why I wondered, in my first post in this thread, whether very much turns on whether mathematics is discovered or invented; to put it more brutally, "who cares?"

It turns out that it depends on how you define "discover" and "invent". I come down on the side of "discover" for the reason that I stated: all the mathematics that a mathematician develops follows inevitably from the axioms which which he begins. You come down on the side of "invent", because we can't see mathematics:

We don't see numbers and mathematical formulae when we observe our world

We're not disagreeing about anything, just using words differently.

Agreed: whether it's philosophizing or filosofeezing about math, I am not sure I'm the best person to be doing it. To me, discover and invent are just words - they don't constrain reality but only attempt to formalize some element of it such that we can communicate a meaning, so if after using them the meaning is still poorly communicated, it's the fault of the words not indicative of some deep truth about the universe.

Both have some bearing, but both seem to mean something they shouldn't properly mean with respect to how math.

As I said, I wasn't commenting on your post, but on that of jamest.

Spearthrower wrote:

Evolving wrote:

Spearthrower wrote:
We could say, for example, that Darwin [i]discovered natural selection - but not that he invented natural selection, despite being (among) the first to alight on the idea and accurately describe the phenomenon.

Indeed, and how very stupid not to have thought of it.

Did you ever see any of the history of authors predating Darwin who clearly understood the concept, but never bothered themselves with attempting to formalize it? A Scottish guy writing a book about growing the best kind of wood for boats, for example.

I knew the idea was in the air, there for the grabbing - hence the quote in my signature, which I was (not very subtly) referencing.

jamest wrote:We don't see numbers and mathematical formulae when we observe our world, we first have to have an idea of what's happening around us and then express that idea using the language of maths/physics. Maths is created to mirror our idea, but we no more discover maths than we discover ideas (about the world) in our minds. Ideas come through creative thinking and even logic is a process of creative thinking, hence maths too is an expression/language of creative thought.

Nothing in the universe is discoverable unless recognised by the mind and the mind only discovers something if it understands that thing, so discoveries in the world actually amount to thoughts in the mind. In effect, what we are observing in the world is a connected series of thoughts about it. The mind's own representation/map of it.

From the onset, minds have invented concepts and ideas to identify and understand. I say 'invented' because concepts and ideas are not observable through telescopes, they emanate from ourselves. In that sense, our minds are the source of all discovery.

ie42