Mathematics is Everywhere Philosophy?
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. I have come to know some folks in NCTM (National Council of Teachers of Mathematics) where there is a pedagogical practice of allowing students to experience mathematics is different locales (often called mathematics walks or mathematics trails). I am on the advisory board of talkSTEM (talkstem.org) which promotes "mathematics everywhere" in terms of video lectures. I have surmised from talking to teachers, that this math walk concept is frequently employed in elementary or middle school, but not in high school and beyond (perhaps the "system" prevents these walks since students have to "study for the tests and exams)". 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).
Comments (36)
Beats me. I was a math prof for years and never had an interest in seeing math in everything. :chin:
[quote=Galileo Galilei]Mathematics is the language with which God has written the universe.[/quote]
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,
[quote=Albert Einstein]Everything that can be counted does not necessarily count; everything that counts cannot necessarily be counted.[/quote]
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
"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?
https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis
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.
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'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.
: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.
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.
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.”
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.
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.
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.
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.
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.
Maybe you learned something about the relationship of mathematics with the world. Maybe not.
Mathematics.
[quote=Wikipedia]Mathematicians seek and use PATTERNS to formulate new conjectures; they resolve the truth or falsity of such by mathematical proof.[/quote]
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):
[quote=Hillary Putnam]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.[/quote]
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.
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.
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.
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.
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)
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 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.
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.
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.
There is very definitely a world outside your mind. The issues is how we understand the relation between the subjective and the objective aspect of experience. There aren’t simply in themselves subjects and in themselves objects colliding with each other. Even Kant knew better than that. We have to understand that what it means to be an object is to play a role in a constructive process that a subject generates in an intersubjective space. And for its part , to be a subject is to be changed in its organization and understanding by the objects it construes. So each side of the equation is changed and shaped by the other.
“Knowledge is taken to consist in a faithful mirroring of a mind-independent reality. It is taken to be of a reality which exists independently of that knowledge, and indeed independently of any thought and experience (Williams 2005, 48). If we want to know true reality, we should aim at describing the way the world is, not just independently of its being believed to be that way, but independently of all the ways in which it happens to present itself to us human beings. An absolute conception would be a dehumanized conception, a conception from which all traces of ourselves had been removed. Nothing would remain that would indicate whose conception it is, how those who form or possess that conception experience the world, and when or where they find themselves in it.
It would be as impersonal, impartial, and objective a picture of the world as we could possibly achieve (Stroud 2000, 30). How are we supposed to reach this conception? Metaphysical realism assumes that everyday experience combines subjective and objective features and that we can reach an objective picture of what the world is really like by stripping away the subjective. It consequently argues that there is a clear distinction to be drawn between the properties things have “in themselves” and the properties which are “projected by us”. Whereas the world of appearance, the world as it is for us in daily life, combines subjective and objective features, science captures the objective world, the world as it is in itself. But to think that science can provide us with an absolute description of reality, that is, a description from a view from nowhere; to think that science is the only road to metaphysical truth, and that science simply mirrors the way in which Nature classifies itself, is – according to Putnam – illusory.
It is an illusion to think that the notions of “object” or “reality” or “world” have any sense outside of and independently of our conceptual schemes (Putnam 1992, 120). Putnam is not denying that there are “external facts”; he even thinks that we can say what they are; but as he writes, “what e cannot say – because it makes no sense – is what the facts are independent of all conceptual choices” (Putnam 1987, 33). We cannot hold all our current beliefs about the world up against the world and somehow measure the degree of correspondence between the two. It is, in other words, nonsensical to suggest that we should try to peel our perceptions and beliefs off the world, as it were, in order to compare them in some direct way with what they are about (Stroud 2000, 27). This is not to say that our conceptual schemes create the world, but as Putnam writes, they don't just mirror it either (Putnam 1978, 1). Ultimately, what we call “reality” is so deeply suffused with mind- and language-dependent structures that it is altogether impossible to make a neat distinction between those parts of our beliefs that reflect the world “in itself” and those parts of our beliefs that simply express “our conceptual contribution.” The very idea that our cognition should be nothing but a re-presentation of something mind-independent consequently has to be abandoned (Putnam 1990, 28, 1981, 54, 1987, 77)
I tend to equate the human science of Mathematics with knowledge of the Logical structure of the universe. In mathematical analysis, we are describing certain logical relationships between things. And one result of those "equations" is a unified & holistic view of otherwise independent parts of reality. The physical parts of reality are visible and tangible. But the web of interrelationships is invisible, except to rational minds. So, Mathematics is essentially a form of Mind-reading, in the sense of Hawking's quote about knowing the mind of God.
With that broader notion in mind, I would call the mathematical aspect of reality : Meta-Physical. That's because it applies, not just to material relationships, but to meaningful & moral human (mental, emotional) relationships. Logical relationships have both numerical values (ratios) and moral values (true/false; good/bad). But those who focus their mathematical investigations on the parts, may not "see" the whole picture, that Hawking referred to as "God". Of course, he was not referring to the god-model of any particular religion, but to the Nature-god (or Logos) of the philosophers, specifically Spinoza. And, in that all-encompassing sense, Mathematics (Logic) is part & parcel of "everything". :smile:
"If we find the answer to that, it would be the ultimate triumph of human reason—for then we would know the mind of God" ___Stephen Hawking
same thing could be said about poetry
which is the opposite of math
Curious. How is poetry the opposite of math? I'm not saying it isn't. But there is poetry in math, usually comprehensible only to those who study the subject.
poetry is like dreams
information without logic
I can accept that.