So, putting all of the pedagogy and math trails aside, what exists within philosophical discourse that promotes this way of seeing? — Paul Fishwick

Beats me. I was a math prof for years and never had an interest in seeing math in everything. :chin:

Mathematics is the language with which God has written the universe. — Galileo Galilei

Comparison (Grammar)

"er" and "est" as in blacker and fastest.

"more" and "less" for a word like "beautiful": more beautiful or less beautiful

So long as comparisons are made and they are made, quantification/numericizjng follows naturally; after all, numbers bring to the table arbitrary levels of precision, something vital to the enterprise of comparison.

What about black holes, gravitational singularities? I hear math breaks down "inside" them. Is this a problem with the scientific theory in question, a call to develop a new branch of math, or is it that black holes are beyond the reach of math? Something to do with infinity?


Then this,

Everything that can be counted does not necessarily count; everything that counts cannot necessarily be counted. — Albert Einstein

Either Einstein was in the early stages of dementia or he was onto something.

I have short argument that math is everywhere.

Definition of compare: same as (=), greater (>), lesser (<)

1. Either you compare or you don't compare (tautology)
2. If you compare then math is everywhere (see definition of compare above)
3. If you don't compare then math is everywhere (everything is same (=))
4. Math is everywhere or math is everywhere (1, 2, 3 CD)
Ergo,
5. Math is everywhere (4, Taut)

QED

I thought it was just me pacing out the central circle of a football pitch toe to heel, circumference then diameter, then spending the rest of the walk caculating pi. Great to hear it is a thing, even if a primary school thing. 'Geometry' is 'measurement of earth'. I expect you know this:

"Scholars of embodiment seek to evaluate the intriguing hypothesis that thought—even thinking about would-be abstract ideas—is inherently modal activity that shares much neural, sensorimotor, phenomenological, and cognitive wherewithal with actual dynamical corporeal being in the world. By this token, higher-order reasoning, such as solving an algebra equation, analyzing a chemical compound, editing a journal manuscript, or engineering a spacecraft, transpires not in some disembodied cerebral space and not as computational procedures processing symbolic propositions but, rather, by operating on, with, and through actual or imagined objects." https://www.frontiersin.org/articles/10.3389/feduc.2020.00147/full

Two historical roots - Kant and Plato:

Definitions as well as fundamental mathematical propositions, for example, that space can only have three dimensions, must be “examined in concreto so that they come to be cognized intuitively”, but such propositions can never be proved since they are not inferred from other propositions
https://plato.stanford.edu/entries/kant-mathematics/

Socrates demonstrates his method of questioning and recollection by questioning a slave boy who works in Meno's house. This house slave is ignorant of geometry. The subsequent discussion shows the slave capable of learning a complicated geometry problem. Socrates, however, argues that the slave could not have learned it from Socrates, since Socrates did nothing but ask him questions...
https://en.wikipedia.org/wiki/Meno%27s_slave

Plato has the wild idea that we know maths from a previous life and only need to unforget what we have already learned. The answer is dubious but the question is great: where do our mathematical intuitions come from?

But it occurs to me that's not what you are asking. You are not asking how our mathematical intutions relate to experience. You are asking how all our experience can be interpreted as mathematics - is that right? But to me that just seems false. I feel outraged when I hear about some of the injustices in the world. No maths involved, as far as I know. But have I understood the question?

Look up Max Tegmark

I was going to say the same thing, or actually, just post this link:

https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis

I appreciate all of these excellent points and suggestions. Let me clarify my interest in the posed question. The "mathematics is everywhere" way of seeing is targeted to mathematics education,
but can easily be extended to abstract mathematical areas of statistics and computer science.
I see it as a means for teaching and learning. I am not suggesting that mathematics is the main,
or only, way of seeing since one can walk around the house or apartment and "see" objects from multiple viewpoints, and not just mathematics. For example, chemistry, physics, design, social issues (e.g "Ways of Seeing" by Berger in art criticism) are all worthy ways of seeing.

Perhaps a more accurate question on my part would be "What are the philosophies of mathematics that underlie the movements in math education based on math trails/walks?"

Some potentials are included in this thread. Tegmark in "Our Mathematical Universe" takes the ontological view that "the universe IS mathematics". As much was said by Galileo and others. And I agree that such views and discourse do contribute an answer to "mathematics everywhere." Some other movements such as digital physics and Wolfram's "new kind of science" seem similar to Tegmark's thesis.

I may be seeking a simple answer to a complicated question. There may be no single philosophy of mathematics that is situated empirically in seeing math in everything. I am not seeing anything that stands out here: https://plato.stanford.edu/entries/philosophy-mathematics/

Phenomenology and Empiricism might also contribute as philosophies as well as embodied mind theories.

There may be no single philosophy of mathematics that is situated empirically in seeing math in everything. — Paul Fishwick

It may be a matter of perspective or personality rather than philosophy. In my years as a math prof I don't recall any colleague particularly interested in seeing math in play around them to any prolonged extent. But there are probably some out there who are like that. However, I'll bet Max Tegmark is not of that ilk!

What you refer to might be ontological, but I doubt it is an aspect of philosophy of mathematics, which
seems to focus on foundations, math systems, logic, Platonic vs non-Platonic, etc. But I could be entirely off base here, and welcome illuminating comments. :cool:

I have searched on and off for years on what philosophical movements promote, or are in agreement with, the idea that everything in our experience can be interpreted/translated as mathematics. — Paul Fishwick

I'd agree with this if one important and logical field of mathematics is taken into account: the uncomputable and the incommeasurable.

Then math explains everything.

The problem is that a lot is there in which you cannot make a function, cannot compute and the only thing you can do with these unique mathematical objects is to use narrative. But, you could argue based on mathematics that seeking functions or algorithms is not so smart thing to do.

Yet if you argue that everything is computable and measurable, then I have to disagree with you.

Beats me. I was a math prof for years and never had an interest in seeing math in everything. :chin: — jgill

:100:

Reminds me of those people who memorize digits of pi. The only people with zero interest in doing that are math majors.

I don't think everything is math. The most important aspects of life are not quantifiable and not subject to logical or rational analysis.

It's hard to do better than a good math teacher and an interested student but probably good to have a mix.
The field trips or outings I remember were more social events and maybe some content was built on later. One ecology trip was to a low head dam. We found out much later they are also known as drowning machines
Hope that was what you were asking about. I'm not an academic so maybe that's a students memory perspective
I do remember back and forth conversations with math teachers and that's what worked for me. Like having explained there is a thing called calculus for the very first time.

Well math studies the simplest possible structures. And even here when something simple enough becomes a little long or strange the problems become formidable, some perhaps even unsolvable.

But the simplest possible thing we can cognize with would have some connection to the world, if only structurally. But beyond that, mathematics is of little use. What can math do for a person who suffers from severe depression? One can always say well, this person's amygdala, or whatever brain part, is a few millimeters too big. Or that the color red is the reflection of light at X frequency.

But it doesn't tell you about the experience of the color red. So, yes, math can be used in many areas of life, in some manner, but I don't think it tells us much about our ordinary experience.

As Bertrand Russell said:

"Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.”

Phenomenology and Empiricism might also contribute as philosophies as well as embodied mind theories. — Paul Fishwick

In a narrow, defined sense the question of "what is mathematics?" is the same question as "what is information?". Neither is physical matter and neither is a stand alone non-physical (noun). Both are neuron contained non-physicals. Philosophy fails to identify this relation and the mathematics profession has failed to identify this relation.

"What are the philosophies of mathematics that underlie the movements in math education based on math trails/walks?" — Paul Fishwick

I went to lunch today with retired colleagues, one of whom was a professor of math education. I asked them about math trails or walks and neither was familiar with the notion. So even academics can draw a blank on the subject.

If there was a philosophical area behind the concept I think it might have shown up here by now.

I think you are right. Thanks for asking them.

It would be great to see education that focused a little more on the creative side of mathematics. You know, there are many now who understand, elaborate, and compute the formulae and concepts, but something in me just wants to see students that have some room to experiment, play, and take side-tracks. It would be great to take some of the pressure off being the next Gauss or Euler, and to just enjoy the beauty of the subject.

The true driving force has never really been physical gains, fame, or notoriety, has it? I always imagine the geniuses as being very much adverse to those types of motives. It might help individuals to ‘see math in everything’ if they saw it as something taken for itself, since in my view it has developed a reputation for being laborious, wretched, and ugly, which just doesn’t fit it right.

So, putting all of the pedagogy and math trails aside, what exists within philosophical discourse that promotes this way of seeing? The closest I have seen in my research is "embodied mathematics" (e.g. Lakoff/Nunez). — Paul Fishwick

I am a civil engineer. There is nothing I ever did that "embodied" math for me like taking surveying in college. Trigonometry, measurement and measurement error, statistics, planar and spherical coordinate systems, mapping, precision. I came out of that class with the sense of holding the world in my hands with mathematical tools. Related ways of studying math in the world - orienteering, map making, drafting, CAD. A lot of this stuff can be done without formal surveying tools.

Beyond that, the class that most helped me see the world in mathematical terms is probability and statistics. When I did statistics, I could feel the universe around me, fighting me like a spinning gyroscope as I tried to turn it . And calculus too. When you do calculus, you can feel how the world changes. I loved partial derivatives - like strings pulling on the world to make it move. Each piece of the world attached with a network of strings.

Sorry - these are not philosophical tools, although they do teach philosophy as much as they teach math. And they teach physics as much as they teach either.

There is nothing I ever did that "embodied" math for me like taking surveying in college. — T Clark

I took a class in surveying at Ga Tech in the 1950s, roaming all over campus with classmates and equipment. Then, the next summer, working for the US Forest Service, I led a three man (student) team through two miles of forest, laying a timber sales line. We started at one end and worked toward the midpoint, then began at the other end working towards that hypothetical point. Coming in the second time we found nothing where there should have been our markers. Not even familiar surroundings. It took four hours of determined scouting to find the initial midpoint. We never told authorities about their smooth continuous survey line being instead pretty jagged in the middle.

My enjoyment of math did not stem from that experience.

My enjoyment of math did not stem from that experience. — jgill

Maybe you learned something about the relationship of mathematics with the world. Maybe not.

Mathematics.

Mathematicians seek and use PATTERNS to formulate new conjectures; they resolve the truth or falsity of such by mathematical proof. — Wikipedia

Relations (Philosophy).

Relations (Math).

Apropose my previous post, mathematics is ultimately about, as the quote says, PATTERNS. All intellectual activities humans engage in seem to be about either 1. detecting patterns and/or 2. explaining these patterns. This includes philosophy too (see below):

It's a tendency in philosophy to look for generalizations [patterns] that covers all the cases and we always lose but we can't resist trying. — Hillary Putnam

So yeah, going by what Hillary Putnam says, philosophy is math and vice versa since both seek patterns (generalizations) but mind the words "...we always lose..." which to my reckoning simply means that the patterns philosophy is interested usually don't cover all the bases i.e. there are exceptions that gum up the works. Does this matter? Well, in one way it doesn't - philosophy and math both seek the same thing (patterns) - and in one way it does - in philosophy, unlike math, patterns are not universal (there are special cases which break the pattern) which when examined closely sounds very much like saying there are no patterns. Hence, though both math and philosophy are pattern-seeking projects, the former finds them with ease I might add while the latter struggles to find even one. Math is everywhere as a goal but not everywhere as an achieved goal.

Good quote for Putnam and for the focus on pattern-seeking. In my classes over the past 5 years, students have been encouraged to interpret things they see on a daily basis as mathematics, or its natural abstract extensions (statistics, computer science). In self-critique, I wonder where this interest originates. It might be based on an artist-workflow where perception is strongly valued and taught in classrooms. So, enter psychology.

psychology — Paul Fishwick

As far as I can tell, psychology basically carries out experimental studies on how people think i.e. their raison ​d'être is to check for generalities/patterns in our thinking.

The psychologist's question: What would all/most people think given such and such?

I'm not sure of how much progress has been made in this regard though. Do most people concur i.e. are there tangible motifs we can then work on or is there too much variability to make even partial generalizations?

The next step as in any scientific endeavor seems a tough nut to crack - coming up with an explanation for observed patterns if any.

That's the math in psychology.

Coming to the psychology in math, I guess the first step is done with - we know we seek patterns. Next step - Why? Well, it helps to know how the world behaves, we can formulate plans - simple ones like going to the places where migrating bison herds can be found or elaborate ones like sending rovers to Mars.

What I meant by adding psychology into the mix was that "seeing math everywhere" requires an attention to what is sensed and in stage two, what is perceived. A focus on doing this, perceiving, seems to be more prominent in certain disciplines. Some math folks with math walks stress it, but many do not as pointed out by jgill. In the arts and in art museums, perception and paying attention to one's surroundings is central. This stress is what I am promoting in understanding mathematics and computing. I am not suggesting everyone do it, but it would seem to correlate to an "arts way" of interpreting.

going by what Hillary Putnam says, philosophy is math and vice versa since both seek patterns (generalizations) but mind the words "...we always lose..." which to my reckoning simply means that the patterns philosophy is interested usually don't cover all the bases i.e. there are exceptions that gum up the works. — TheMadFool

Post structuralist philosophy is based on the idea that what gives a pattern its meaning is it’s difference from a previous pattern This difference both defines the pattern and reminds us that patterns always depend on something outside of themselves in order for them to be what they are. Their condition of possibility is the flow of time and history. One could liken this to a hegelian dialectical movement , with the difference being that dialectic makes the flow of history itself into a logical pattern. Post structuralism, by contrast, sees the transition from pattern to pattern as not capturable in any logic. Even without. stable patterns themselves (cultural, empirical) we find this incessant movement , and this makes math and logic tricks that we use to ‘freeze’ the incessantly transformative movement of experience into abstracti objects and forms. To see math everywhere is to pay attention to a second order derived action that we perform that covers over its basis in living.

To see math everywhere is to pay attention to a second order derived action that we perform that covers over its basis in living. — Joshs

If I understand what you're saying (and it wasn't till the very end that I thought I did), existence is what it is, and math is a secondary thing that humans use to model and explain it. In which case "math is everywhere" spoken by humans, is in the same sense that "echoes are everywhere" is to bats. Which is to say, "math is everywhere" is purely a human-centric conceit. Math is nowhere at all, except in the mind of humans. Which I believe, on my formalist days. And disbelieve, on my Platonic days. After all 5 is prime, and it's hard to argue that it would be false if there were no people around.

If I understand what you're saying (and it wasn't till the very end that I thought I did), existence is what it is, and math is a secondary thing that humans use to model and explain it. In which case "math is everywhere" spoken by humans, is in the same sense that "echoes are everywhere" is to bats. Which is to say, "math is everywhere" is purely a human-centric conceit. Math is nowhere at all, except in the mind of humans. — fishfry

I didn’t mean to distinguish between the world for us as humans and the world as it supposedly is in itself. This distinction belongs to the abstracting act that makes math possible. Math and formal logic evolved along with the concept of the external object. Each implies the other, and both are abstractions from our pragmatic engagement with the world. We are always pragmatically involved with things. Things matter to us, are significant to us in relation to our concerns and goals. Out of these contexts of relevance , we abstract what we call empirical objects which supposedly exist in themselves, apart fromour interaction with them and the purposes for which we are involved with them. This abstracting and separating off of an external world from pragmatic subject-object engagement makes mathematics and formallogic possible , but at the expense of losing sight of the pragmatic contexts which not only generate mathematical and logical objects but give them their meaning. Math is everywhere is the same as saying empirical objects are all around us , as if we are just one object among the furniture of the world. But fundamentally, the idea of a world of things existing independently of us is incoherent.

“… we can see historically how the concept of nature as physical being got constructed in an objectivist way, while at the same time we can begin to conceive of the possibility of a different kind of construction that would be post-physicalist and post-dualist–that is, beyond the divide between the “mental” (understood as not conceptually involving the physical) and the “physical” (understood as not conceptually involving the mental)….
natural objects and properties are not intrinsically identifiable; they are identifiable only in relation to the ‘conceptual imputations' of intersubjective experience.”
(Evan Thompson)

But fundamentally, the idea of a world of things existing independently of us is incoherent. — Joshs

Not a naive realist then. But surely the world didn't come into existence when you were born. Or when the first fish crawled out of the ocean (or whatever they did, I'm not a biologist).

I am out of my depth chatting with someone who drops the phrase Hegelian dialectic. I didn't actually understand anything you wrote. I should stay out of this. Except to say that for what it's worth, and for sake of discussion, I'd be perfectly happy to defend the thesis that "Math is nowhere." On my formalist days, of course.

But surely the world didn't come into existence when you were born. Or when the first fish crawled out of the ocean (or whatever they did, I'm not a biologist). — fishfry

But when we model the world we’re not capturing it in a bottle, we’re interacting with it, making changes in it for our purposes. I know this seems counterintuitive. For centuries we assumed that the world is a set of object out there and our job is to mirror it with our representations.
But when we know something we are engaged in an activity involving that thing, transforming that thing in a certain way. Perceptual psychologists discovered this about the way that we perceive our perceptual world. To perceive something is not a passive inputting of a stimulus. It is a constructive activity involving anticipating of the way the world will respond to our behaviors in relation to it.
Looked at this way, the evolution of knowledge isn’t getting closer and closer to something sitting static out there. It’s the building of something always new, in conformity with our changing needs and purposes. At each step the ‘outside’ world only announces itself as affordances and constraints intricately responsive to our creative efforts.
Math and logic are a part of this but are only one element in a dance that moves back and forth between the fixing of set patterns and their dismantling and reformation as fresh structures.

But when we model the world we’re not capturing it in a bottle, — Joshs

None of what you wrote convinces me that there's no world out there. Except on the days that I'm certain I'm a Boltzmann brain. And even then, there is a world outside my mind.

But when we model the world we’re not capturing it in a bottle, we’re interacting with it, making changes in it for our purposes. I know this seems counterintuitive. For centuries we assumed that the world is a set of object out there and our job is to mirror it with our representations.
But when we know something we are engaged in an activity involving that thing, producing that thing. Perceptual psychologists discovered this about the way that we perceive our perceptual world. To perceive something is not a passive inputting of a stimulus. It is a constructive activity involving anticipating of the way the world will respond to our behaviors in relation to it.
Looked at this way, the evolution of knowledge isn’t getting closer and closer to something sitting static out there. It’s the building of something always new, in conformity with our changing needs and purposes. At each step the ‘outside’ world only announces itself as affordances and constraints intricately responsive to our creative efforts.
Math and logic are a part of this but are only one element in a dance that moves back and forth between the fixing of set patterns and their dismantling and reformation as fresh structures.

Excellent @Joshs
More philosophers need to really understand and engage what you have written above and your next quote here;

"But fundamentally, the idea of a world of things existing independently of us is incoherent."— Joshs

Yes the material world is seperate from us,but it does not exist independent of us.

There are religious and "eternity" conclusions to be gleaned from these facts as well.