Clarifying position on learning/knowing mathematics

I attempted to post this as a comment at @blaw0013 's post from 2012 but it was causing me all sorts of issues since what I wrote was too long... and so I decided to post it instead and invite conversation

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If all interactions are inherently coercive, and in my mind similar to gravity, I would like to add my own mass in here and invite you to think with me on a few things.  Before I go on, I wanted to clarify that I do not have a set position on this yet, and am hesitant in stating that I have decided on a position.  In fact, I am unsure that I would agree with the idea of taking a position - unless it is my own dynamic position that I have subjectively decided on.

First, let me begin by providing some context on how I got here.  Bryan Meyer and I have been conversing over DM (twitter was a poor medium for in depth discussions) for a while about a variety of different topics.  More recently I asked about how he has identified his position with respect to learning and knowing.  To this, he responded that he identifies with forms of radical constructivism, and stated that the "social" was accounted for through the idea of intersubjectivity.  A few more back-and-forths later, he asked if I thought of mathematics as invented or discovered (yes, we were still conversing through twitter at this point... silly us).

Looking back, I think my answer would have differed if he had asked whether I thought the "learning and knowing" of mathematics was invented or discovered.  Regardless, this is what I said:

"Here are my current thoughts on that: I think it's both.  I think mathematics is individually invented, but collectively discovered.  With respect to mathematics as a subject, I like Shapiro's view of structuralism, where mathematics exist as structural relationships. To make sense of ideas in mathematics, one must "invent" concepts from being situated with the rest of existing knowledge, or resolving cognitive dissonance.  At the same time, mathematics as a social practice is discovered collectively.  If mathematics is a landscape of buildings, then individuals are inventing different parts of the buildings on their own, but others are inventing the exact same things.  The superimposed image of mathematics is the structural relationship that is discovered."

We left this a bit (because we were still on twitter and it was difficult to flesh everything out) and we went on to having him describe his personal epistemology.  To this he stated that no knowledge is external to the knower, and therefore it is invented.

I responded with:

"...back to my thoughts on how mathematics is "individually invented, but collectively discovered."  I am basically attempting to make a case for a duality between invention and discovery.  Mathematics as something to be individually constructed in the process of sense making -- which I think is where you are coming from, and I agree with this.  On the other hand, I believe that there is an existence of mathematical properties prior to my construction of concepts.  e.g. the concept of addition. My thinking is that in the process of our individually making sense of addition (for ourselves), this is a kind of collective discovery of some sort of structural property of mathematical objects - in the sense of e.g. Shapiro's concept of structuralism. But I am not set in stone with respect to my personal epistemologies, learning theories, and theories about the nature of mathematics...I think your question is - then how is it possibly inventing if it already exists?  I am currently reading about the phenomenology of practice which I think may have had an influence on my current thinking.  Imagine the concept of a bridge.  Imagine an individual who has no knowledge of this concept.  As this individual becomes familiar with the concept of a bridge - in whatever way they choose to define bridge - they are constructing their own idea of what a bridge is - in relation to the rest of what they understand. And this act is invention to me -the construction of a concept with relation to an existing body of knowledge that the individual has constructed so far... I am unsure if you have seen the movie "the gods must be crazy," but, the idea is hat a coke bottle falls from a plane to a village, and then the villagers began to use the bottle for things outside of what the bottle is typically used for in our culture.  The development of what it is and what it is used for, would be invention. However, at the same time, there is pre-existing properties of the bottle that allows it to tend toward certain inventions. The process of understanding and identifying these properties - would be discovery."

After a few more exchanges, we recently moved onto e-mail, and his post (and your response here) was brought up, along with a few images of quotes from von Glaserfeld's (1996) book on Radical Constructivism (studies in mathematical education).

Ok, and now that we're in the present, I hope the above has invited you enough to our conversation!  Feel free to comment on my existing analogies and half-baked ideas.

First, let me address a wondering from this post & comments.

"I would argue that as soon as you ask the question, you have created an unjust space between people--there is no way that one could be deemed not correct, nor more correct. Each is assumed to be "correct," i.e. viable, for that other autonomous being."

I wonder about the degree of justification for "viability."  Even if we stand on the platform of individualistic ways of knowing, is it not still possible to identify degrees of "correctness" to the rest of his/her own networks of constructions?  As a simple (and perhaps ill-formed) example, if I believe in 1+1=2, then if I state that 1+2=2 I would have a less degree of "correctness" to someone who believes 1+2=2, and then state 1+2+1+2=4?  My example is flawed in many ways, but I hope I am somewhat illustrating my wondering.

Now I'd like to return to the attempt of clarifying my own tentative position (and in fact without this act of doing so, I would have no position to speak of).

On mathematics:

As I attempted to illustrate through the copy-and-pasted DM conversation, I currently envision mathematics as invented and collectively discovered.  I think my current state of opinion has to do with a combination of different experiences.  Thinking back, in an effort to clarify, I think the aspect that I believe is "invented" is the learning of mathematics.  The aspect that I believe is "collective discovered" is the nature of mathematics as structural relationships. 

What I am saying here is actually related to my wondering earlier.  Let me attempt another example.  Let's say I am an individual who has constructed the concept of 1 as a lone object with several properties.  This makes sense to me as it relates to my other constructions through my interactions with the world.  As I construct a new concept of 2, I relate to my previous constructions for the concepts of 1, as well as all of its innate properties of one-ness.  By doing so, improving my own "correctness" for the concept of 1.  The structural relationship that I have "invented" would then be that 1+1=2.

At the same time, I arrived at this conclusion not only based on my own constructions of other concepts, but also relying on the inherent properties of one-ness and two-ness.

I am unsure if that example was any good...  I suppose this relates to my tentative thinking that while abstract mathematical "objects" do not exist, the structural relationships amongst objects - do.

Thoughts? Comments?  This is all getting very foggy for me now haha.

*note* This whole discussion actually is reminiscent of when I was studying the various philosophies of mathematics (which I am unsure I have decided on a strongly set position there either).