Here's a great video pondering the objective reality of mathematics, and running down all the different schools of thought on where mathematical truth comes from -- does it exist outside of systems of codification by intelligent beings, as an eternal part of the universe; or is it something that we invent through codification?

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Math is good and well but it’s not a patch on Maths.

The reason Europe has better education is because they buy their math in bulk.

I thought it was the Hylaean Theoric World?

Did you just perform a hypertrochian transquaestiation?

You two are giving me traumatic flashbacks.

The correct response being “Hey, not so fast!”

The slines aren’t going to get this.

I preferred the downwick version! Plato’s Ideal World of Forms thank you very much!

This is like required viewing for stoned college freshmen.

My guess is that he flunked out of philosophy and never got over it.

Gee, my guess is that he throughly enjoyed philosophy.

I once tried to construct the basic four operations using solely union and intersection. I haven’t trusted multiplication or division since.

For anyone with an interest in cognitive science, Lakoff and Nuñez’s Where Mathematics Comes From is quite a read. Well worth your time, even if by the end you still find yourself disagreeing with them.

Did you at least allow yourself recursive definitions and set membership?

{} = 0, { {} } = 1, { {{}}, {} } = 2, etc.

A + 0 = A
A + 1 = { A } ∪ A
A + (B + 1) = (A + 1) + B

That should be enough to get you addition and subtraction.

A * 0 = 0
A * 1 = A
A * (B + 1) = A * B + A

And there’s multiplication and division.

I’d like to say that I was just being excessively strict but really I was just incredibly naive. I didn’t even think to define the operand (in this case, the natural numbers).

This is not news. Aristotle wrote, in the earliest writing on mathematical philosophy, that numbers and the way we use them were nothing more than a social construction.

That said, if by “not real” you mean “social construction,” then pretty much almost everything humans have ever done is “not real.”

Plato wrote on mathematical philosophy before Aristotle, and of course he thought that mathematical truths were eternal, part of the realm of forms (the belief that math has some sort of objective reality is still known as “mathematical platonism”). And all human knowledge (including, say, “the Earth is round”) is of course a “social construction” in the broad sense that the process of humans coming to agreement about some fact necessarily involves social interactions, but hopefully you agree this doesn’t discredit the idea that there is such a thing as objective truth (in math or any other subject).

but hopefully you agree this doesn’t discredit the idea that there is
such a thing as objective truth (in math or any other subject).

Why “hopefully”? While I think the idea of objective truth has a lot going for it I also think the idea that “objective truth” is self-contradictory has a lot going for it too. Why do we have to pretend to know the unknowable?

“Hopefully” because I think this would be a bad argument for showing that there can be no objective truth–I wasn’t expressing any hopes one way or another about whether you believe in objective truth or are agnostic/disbelieving, just commenting on this specific argument which some people might make.

“This is not news.”

Who said it was?

Also, the fact that this is not news is not news.

Not sure what hurt my head more…the ideas discussed or all that rapid fire editing!!

1. Now my brain hurts.

2. Is this guy related to Jason Lee in any way? Because his mannerisms and the rhythms of his speech really remind me of Jason Lee.

The unit exists. How many children do you have? One? Many? More than one? Not many? All set based concepts surrounding the concept of a single unit. Is the argument REALLY about how we convert the theory of units into a language? I don’t think it matters if we smell units, taste units, sing units, describe units as colors, or fart units. The undeniable fact that units exist and the study of units leads to an elegant language still remains. Just like grunting led to Shakespeare. However the details turned out, math was inevitable. Saying math may be a fictional creation is the same as saying sentience may be in our heads. What good does that kind of thinking do us.

There’s some condensed chunks of energy, a bunch of atoms, occupying a tiny bit of space in an incredibly small piece of their coincident progress along some entropic journey, commonly called ‘time’. And this is what you arbitrarily decide to call a child? Maybe those bits of stuff have better things to do.

“some…bunch…tiny bit…incredibly small…” – try to express our idea without any quantification or assumption of any form of unit.

There’s condensed chunks of energy, atoms, occupying space in a piece of their coincident progress along some entropic journey, commonly called ‘time’.

” chunk”. You can’t do it. We are slaves to the time stream.

If you take “chunks of” out of there, then, it’s devoid of unit or quantification. and still sensical.

I don’t think it’s an undeniable fact that units exist. In fact, this video put into language for me a world of thought I had been thinking, and having a hard time explaining to my friends for the last 15 years. I basically agree that if One exists (what you call The Unit), then all of Math exists. But I think that One-ness – like the circle – is an illusion that the human brain perceives because it is built to recognize patterns. (FYI, there are no circles in nature or reality. The human brain just forces you to perceive circle-like things as circles, and I posit that it does that same to things that appear unitary.)

You’re like, “Of course discrete units exist! Do I have one child, or do I have two?!?” And my reply is along the lines of the adage that “You can’t step into the same river twice.” Your one child is never, moment by moment, the same child that it was before its last big meal. We only perceive it as such because the human mind – mere software – breaks off these huge chunks of the universe and and forces things into the pattern of one-ness. Even at the strictly molecular level, atoms do not behave as discrete units – so as a predictive model, unitary mathematics do a shitty job of predicting reality.

But even though circles do not exist anywhere in nature or reality, we drive around in cars with round wheels that conform to our idea of what is “circular.” And so it is with One-ness or unitary-ness — our whole existence and experience depends on seeing it. An amoeba’s existence, not so much.

And we create equivalancy operations in which Mary and John are “equal” in a way that they can be “added” together to have two children.

Hmmm…you have me thinking about the time component of the unit. I have to agree that adds a twist. It’s very anthropomorphic of me to assume every sentient being co-exists with time in the same way we do. Really good point.

Regarding the circle. It’s just a variation on radial symmetry ( starfish) or the natural occurrence of the arc (Nautilus shell). Nature has MANY examples of the radial entity. Even on the molecular level atoms tend to arrange themselves in geometric patterns. So, I’m not so sure about that one.

“What good does that kind of thinking do us.”

Acknowledging that created concepts are not objective truths is a really, really important kind of thinking, and failing at that thinking has caused a hell of a lot of suffering and wrong-headedness throughout human history.

Also, ‘the unit’ exists as a social construction, which is why the term is as fluid as you describe. It doesn’t exist in any objective concrete way, it exists because I decide to delineate a unit. Which is why, say, my son could be a ‘unit,’ but from another subjective frame of reference, the family that contains my son could be a ‘unit.’ Or each cell that comprises my son could be a unit. etc.

I MIGHT have agreed with you up until the new tunneling quantum electron microscopes showed the atom in all its glory. The universe is indeed broken down into discrete units. The only way to say it isn’t is to say you exist outside of our time and space. At that point, we are waxing philosophic with no usable outcome for our efforts.

Oh, okay, Plato, so the atom is an indivisible monadic unit, now?

Posidonius, I think. Nope, Wikipedia says Leucippus (with mad props to Democritus the Mocker). But definitely not Plato!

I was just glibly dating his atomic model to Ancient Greece with an easily parsable name, rather than linking him explicitly to a specific thinker. Though Plato’s “simple bodies” aren’t terribly dissimilar to FrankenPC’s apparent understanding of the atom…

You are staring at the problem as you make your comments. There is perceived discreteness all around us. You’re just trying to cloud the issue with philosophical discourse. Have fun! Gained anything?

Acknowledging ambiguity isn’t clouding anything. Ignoring it, however, is.

You do know that atoms are largely empty space right? Quarks make up protons and neutrons, and quarks themselves are actually a quantum of energy. (wave-particle duality)
You might be able to argue that a quantum of energy is a discrete unit, but since it’s a wave, it’s sinusoidal, which is continuous. I.e., no discrete units!

Atoms are not discrete, they are made up of electrons and quarks. Electrons and quarks are not discrete; as best as physicists can figure out, they are probability distributions or at least act like them.

The unit exists, sure. Your perception is what breaks it up, like the way you can see the individual threads in a shirt if you have good vision, but it’s still a shirt. Maybe you perceive that you have many children, but to my eyes it’s still just your one family, and we’re both right from our own perspectives.

While the meat engine that makes up your own piece of the unit has the emergent property of a subjectively independent consciousness, it’s still part of the divine whole. I do believe I will go home and drink a beer in celebration of that (it’s important to have beliefs, you know). ^_^

Your perception breaks up the unit, sure, but it also forms it. a shirt is one shirt, a hundred threads, 50 threads of cotton with 50 threads of wool and some grams of colorants, a billion atoms and so on. who is to decide, objectively, which is the unit.

Besides our math is based on the biological limitations of our perception of reality. Its been speculated that being with more blurred sight (or senes) would have a less discret math.

Is math a truth about the universe or is math a fiction that comes out of our heads which is part of the universe so if the fiction of math is describing a consistency in out heads so then it’s a consistency about the universe and therefore a truth about the universe? I think I’m having a problem with the word “fiction”.

Would you agree that colour is a fiction? In that while there is a wavelength of light at 650 nm, the only thing that makes it ‘red’ is our perception of it. Nothing about that wavelength is intrinsically ‘red’.

How about the left- and right-wings of politics? Are they real or just fictional?

By the same token, even though mathematics seems to happen in the universe, we can’t break it down and point to the ‘mathematics particle’. It just seems that mathematics happens to be a model that exists beside the ‘real world’ and happens to describe it. At least, that’s the argument.

Personally, I tend to think that there’s something deeper happening and that mathematics is happening in some real way in reality. The reason for this is that purely mathematical applications of physics can predict things before we observe them – thus suggesting that they mathematics has been happening all along and we’ve just uncovered it.Of course, then there’s Godel’s Incompleteness Theorem – which the video touched upon when mentioning that there mathematical concepts that we are not capable of discovering. Godel’s Incompleteness Theorem states that there’s things which are mathematically true, but which can never be proven (no matter how complex the mathematical system used). (I’m massively simplifying because I’m not a mathematician.) Long story short – while mathematics seems to tell us things that are end up be true/real, it can’t tell us everything that’s true/real. Ever. No matter how good we get at it.

By the way for anyone that’s interested in this stuff, I think there are two books that are incredibly useful for informing further discussion:
1. Godel Escher Bach: An Eternal Golden Braid by Douglass Hofstadter. A non-fiction book with regular fictional interludes. Not necessarily the most accessible book, but everyone who I’ve ever met who’s read it has said it changed the way they think.
2. Anathem by Neale Stephenson. A fiction book with occasional non-fiction interludes. ‘Scififantasy’ has already made reference to it in this discussion, but a key component of the book is discussion of the concept of abstract ideas as being real things coming from somewhere.

I would urge anyone who’s had their mind blown by this stuff to read these books.

>Would you agree that colour is a fiction? In that while there is a wavelength of light at 650 nm, the only thing that makes it ‘red’ is our perception of it. Nothing about that wavelength is intrinsically ‘red’.
Perceiving color doesn’t seem anything like fiction. We perceive color fairly reliably to the point that we eventually were able to investigate it scientifically and recognize that the perception of red is a result of our eye seeing that wavelength of light. Fiction would at best be an analogy, and I don’t see how perception and fiction are truly analogous. Perception is the somewhat unreliable way sense organs discover sense objects, fiction is a deliberately false thing told to deceive, edify, or entertain.

The “red” part doesn’t exist anywhere but in our own heads as should be obvious if you think about color blindness for all of 20 seconds.

Existing in our head and fiction are two different things. Correct logical reasoning exists in our head but it isn’t fiction. There’s nothing obvious to me about the idea that perceptions are fictions.

You’re on the right track, now follow it! There are experiments you can do yourself that will prove to you that your perception of red is independent of the wavelength of light… and it’s fun, too.

Try this link and if you think it’s a video hoax, print out a copy of the Adelson Checkerboard and use a pair of scissors to prove it conclusively to yourself. What you perceive to be two different colors can often be exactly the same recieved wavelengths of light, or vice versa, and it’s a very complex cognitive process that creates the color in your mind.

Color understood as qualia, is the subjective experience of sensory stimuli in consciousness conditioned by a lot of unconscious cognition (typically to speed processing up or filter less critical data). Objectively, we now have some understanding of how those qualia are formed based on eyes (or other sense organs) experiencing wavelengths with cognitive filters applied to sense data, and know that both the organs and the innate cognitive filters can be tricked at times. Illusions take advantage of these limits. That still doesn’t make qualia a fiction though, does it? It just means that senses can be unreliable.

Either “when a wavelength of X frequency strikes the eyeball of a normal healthy human s/he sees the color Y” is a true statement or it is not.

It is not true. This is easy to prove.

Therefore, any conclusions that rely axiomatically on color being determined by wavelength of light are unproven. They are based on a false axiom. That’s the main point I was making.

However, if you consider things that only exist in your head to be fictions, then all qualia must be fictitious. A bit different argument though innit?

There’s an organ in your brain the size of a kidney bean that determines your perception of color. The actual received wavelength of light has phenomenally little to do with this – if you spend four hours in a room with only green lighting, when you come out all your perception of color will have shifted. Read Oliver Sachs to learn more about how the idea that wavelengths = colors is fundamentally flawed. Colors are 100% created in the mind and it’s fairly easy to prove this, since they are perceived relative to the mind’s experiences and not directly from what is presented to the eye at the time of perception.

Second the recommendation of GED and Anathem, although I found the latter a little dry.

Very true.

Color is also dependent in the anatomy of the eye, depending on the amount and types of cones and canes in the retina.
If you wish to exclude colorblind people because they are not healthy or out of the (statistical) norm you still have to deal wit dogs that are unable to see the red spectrum of light, and zephyr falcons who can see ultra violet light.

“construct” is probably a better word than “fiction” (particularly since the word fiction typically implies a narrative).

But neither seems appropriate. I think nemomen had it right, it’s really perception. Both contruct and fiction seem to indicate “made-up” but really something that can be created in the mind’s eye can be interpreted many different ways. Perception is based on biology and (as long as your ocular instruments are working correctly) colors register the same.

Perception is based on biology and sociology acording to construtivist theory. In some east asian languajes blue and green are hues of the same color.

Math helps explain what something is and what something does but math is not the something itself.

That’s about as useful as saying “music tells us what notes to play and what effects they have, but music is not the notes.” Yes, it sort of is, if it’s to have any meaningful definition at all.

Is the word the thing? The word ‘tree’ is not the tree itself. So the musical notes written out isn’t the music but a description.

I think we’re all OK on the difference between description and referent; we’re discussing the nature of the thing itself, not the words we define with.

(I didn’t mean “musical notes written out”, or I would have said “score”. I meant “musical notes”, played as music. The music is a thing in itself, real even when not being played, which the score notates. Likewise, maths is a real thing in itself, an abstraction that corresponds to an actual property of the universe, even when it’s not being used to model physical objects.)

So you’re drawing a completely different distinction there – true, but not the point I was getting at.

The notes ARE NOT music though.
I run into so many Jazz geeks that know things, scales, names, transcribed solos, they have resonators inside their horns, they have experience and knowledge and can play many notes well BUT . . .
the notes is not the music.

We have multiple scientific, empirically-deduced physical models for the universe, which are nonetheless contradictory systems. There is the thing, and there is the model of the thing. Mathematics is a model, a semantic system for describing and organizing various systems, patterns, and concepts. A completely alien entity could, through observing and existing in the same universe, in theory come up with a completely different system for describing the same stuff (just as relativistic and quantum models of physics both describe the movement of bodies in the same universe, or how two different spoken languages both symbolically articulate sentient beings’ lived experience of the same universe).

A completely alien entity could also construct a completely different schema for describing, cataloging, and reproducing different frequencies of vibration travelling though gas. Their system would not necessarily be ‘music’ though it would be organizing the same set of phenomena that our ‘music’ does.

I have occasionally tried to speculate about beings whose existence is alien enough that what to us is simple arithmetic is to them complicated higher math and vice versa. Perhaps ones that exist as standing waves deep within a star.

Please, don’t give the Texas Schoolbook Commission any ideas.

I think the reality or fiction of mathematics as a whole doesn’t really make sense to talk about, because some concepts in mathematics are discovered realities, and some are invented fictions. You have to ask about individual concepts on a case by case basis.

For example, it seems like it is a discovered reality that the number of diameters of a circle it takes to measure out that circle’s circumference is about 3.14159 – the mathematical concept of pi. But other mathematical concepts, say the use of zero as a placeholder when writing decimal digits for example, didn’t come from any observation of nature, and you can’t really say they are “discovered”. They are invented techniques.

So it seems to me like you can’t just conflate all mathematical concepts together and say “it’s all natural reality” or “it’s all fiction”. Mathematics has both.

I think that’s a bit of a false dichotomy. Sure, the notation is fiction, i.e. constructed for human convenience. But to point that out you compared apples and cars – the use of zero as a placeholder is a constructed convention, but the existence of zero isn’t, any more than the existence of pi.

The set-theoretic definition of zero is constructed – but it was constructed for the purpose of formalising the observed properties of zero. I think any mathematician would say maths is discovered; only the tools of maths are invented.

“…the use of zero as a placeholder is a constructed convention, but the existence of zero isn’t…”

Okay, you’ve just made an assertion. Back it up.

Fair enough; it’s certainly an assertion. I didn’t back it up because it felt like a fairly obvious one, just like the existence of “one”.

(Summary form only, because I can’t type much more today; somebody expand if need be.)

Cardinal numbers are innately accessible properties of the universe, as abstractions that apply to things in general. I can count three apples on my desk, and “threeness” is an actual property of the universe – cultures with no communication basis or common language can still come up with “three” independently, because they’ll both observe the same abstraction applying to different sets of three things.
If I throw away two apples, I’ll have “one” left, and that’s another observable abstract property. (We can’t ever see “onenesss”, but we can observe it easily and frequently.)

Eat that one, and I’ll have “zero” apples, just as I currently have “zero” orbital mind control satellites.

There are few human languages that don’t have word for this, because absence is a common and readily observable abstract property of objects. Once you’ve understood counting, absence, and subtraction, it’s not a great cognitive leap to realise that zero is a naturally-occurring number, just as three is. (Young children don’t automatically understand this, any more than they do quantity in general, but they learn it very quickly at the same time as they pick up other quantities.)

The notational system that uses zero as a placeholder is a clever human invention – but the use of zero corresponds to the universe in a deep way; the invention made properties of the universe easier to understand and record. (Note that post-zero base-n number systems are used because they’rer much more elegant than those that don’t – they express fundamental arithmetic and algebraic concepts cleanly where cruder systems modelled them much less well.)

So: Our mathematical concept of “zero” is a notational and logical formalism, built by us as a tool, but it’s a representational tool, standing for an observable abstract property, in the same way that a stave stands for a piece of music, or “one” stands for an abstract set cardinality shared by the set of apples I’m currently eating.

The existence of ‘one’ isn’t obvious though, that’s kind of the point of the whole video :)

His point is that you can, indeed view the existance of pi, or zero or even one as a fiction. The more math that you take, the more “abstract” it becomes. Eventually you circle back and realize that even “simple” math is an abstraction.

Oh, I understand his point, I just don’t think it’s true.

Yes, the more you understand maths the more you understand that the mathematical structure we use is an artificially created abstract logical formalism. (By the time you understand set theory, you can understand that “1+1=2” encodes a fantastically baroque set of chained formalism building from vastly simpler operations.)

But by the time you’re at that level, you should also understand that the structure is not arbitrarily created, but has been designed and used precisely because it models “real-world” mathematical properties well.

It gets trickier when we start to mess with the rules and model accuracy on purpose… and interesting how often that still turns out to be a good model for real properties.(Non-Eulidean geometry is the obvious non-mathematician’s example.)

As others have alluded to, like many conundrums of this sort, then answer to the question “does math ‘exist’.” depends on one’s definition of existence. You don’t have to fall down some “philosophers misinterpret quantum mechanics and conclude that our perceptions CREATE reality” rabbit hole to say that if NUMBERS don’t “exist,” than SPECIES don’t either. They’re both abstractions used to describe the objective world. But there’s certainly no obvious place to draw that line WITHIN mathematics. If the whole numbers are real properties of the universe, because we use them to count apples, than so are complex numbers when we use them to describe electromagnetic fields.

It’s maths. Maths, maths, maths maths, maths. With an S.

Was he indirectly riffing on Voltaire: If maths didn’t exist, we’d need to invent it.

And anthropomorphic deities.

Do you really expect the Empire to take language advice from one of our client states?

Why not mathematics over either choice? How does shortening it to maths or math make any kind of sense? How would you shorten other studies? Physics? Politics? Lingustics?

Physics: Physical studies.
Politics: Political studies.
Maths: Mathematical studies.

The pattern you present is inconsistent in its transformations. Adjusting for consistency, you get either:

Physics: Physical studies.
Politics: Political studies.
Mathematics: Mathematical studies.

-or-

Physes: Physical studies.
Pols: Political studies.
Maths: Mathematical studies.

Honestly, all you show is that ‘Mathematics’ is an acceptable shortening of “Mathematical Studies,” which does nothing to address the actual point of contention of whether “Math” or “Maths” is an adequate shortening of “Mathematics.”

I don’t actually agree with your assessment, but I’ll admit that I didn’t check any linguistic sources to provide those examples, and I agree they’re not really analogous.

Just thinking out loud.

“Otherwise, it sounds like you’ve got a lisp.”
Or it sounds like you come from a different country than tw1515tw, the narcissistic solipsist who thinks the entire world speaks exactly the same language s/he does on her/his small island nation.

Forget discussing whether Mathematics is a construct or an objective truth, we apparently need a video to explain to dullards that spoken language is a construct and not an objective truth.

Do you also flip your shit when a Russian YouTube video says “математика” instead of “maths”? Did you really never learn that sometimes people from different parts of the world use different words, or different forms of similar roots, to describe the same object or concept?

Fucking math, how does that work?

This is not going to help me on my Chemistry test.

You should be studying … or at least working on a study calendar

OK, I’m sold. Click Subscribe.

For whatever definition of existence one chooses, I don’t really see how it’s productive to argue about whether math “exists” or not. All I know is that, given enough time, the probability of two ignorant but curious people to develop their own maths that are interchangeable seems to approach 1.

That is to say, math is essentially the same no matter who studies it and develops the mathematical systems. At least with the simple operations. 1+1 will always be 2, unless it’s 10, but base systems are still interchangeable anyway, so there’s still no difference. You can point to one thing, stick another thing next to it, and there will then be two things. And it happens every time. That sounds like to me like a discovery, not an invention.

Math is the best way to describe the universe around us. So unless there’s something better than math for understanding the world, I’m not convinced by any argument attempting to devalue its usefulness and our need for it.

Soooo… if an undisclosed amount of trees fall in the woods, and nobody is counting them, how much noise does it make?

It makes a barking lot of noise

“Does math exist in the universe or does it come from the human [mind]?” If you take the view that mind and matter are not actually different types, then math can be just spirit on another level, every bit as real as the universal one. This is objective idealism. I think that it makes the whole question drop away rather nicely!

Sort of… the problem is that it makes a lot of other interesting and useful questions go away as well.

As the mind consists of reasonable and evolutionarily necessary reversals of perspective away from the knowledge which the body and brain have of themselves and cosmology (ie the mind has a sense of meaning which is both complementary to and in opposition to that of the brain and body) in order to develop and refine tool making capacity maths is both real and unreal – like God. Really.

Pi seems to be inherent. I’m not a mathematician, but I can’t see how you could have any space with more than two dimensions without pi being the ratio of a circles circumference to it’s diameter. e is similarly constrained, and would show up anywhere you had rates of change that depended on one another.

Oh boy, do I have some bad news for you :)

There are plenty of “spaces” (in the formal mathematical sense) where pi has a totally different value than 3.1415…. The fact that it is that exact number is purely an artifact of us thinking that Euclidian geometry is the only “real” geometry.

Take, for instance, the so-called “Manhattan geometry”. Imagine a geometry where instead of being able to go anywhere, you could only go along the lines of a grid (like a car on the Manhattan street grid). Since a circle is defined as all the points at a certain distance (the radius) from the center of the circle, a circle in Manhattan geometry looks like this: http://en.wikipedia.org/wiki/File:TaxicabGeometryCircle.svg

I.e. a circle in Manhattan geometry looks like a diamond. For a radius r, the diameter is 2r and the length of the circumference is 8r, therefore the value of pi in Manhattan geometry is (exactly) 4.

But that’s not the real world, you say. The real world doesn’t use Manhattan geometry! In the real world, if you have a wheel, the length of the edge of the wheel divided by the diameter of the wheel will be (very close to) 3.1415….

No! Not necessarily true! Lets make use of one of the wonderful results of special relativity, which says that an object that is in motion contracts in length relative to a stationary observer (this is called “length contraction”, look it up on wikipedia if you want). That means that if the wheel is spinning very fast, from the perspective of the person standing still in the center, the moving outer edge is contracting in length, while the radius stays the same. In other words, for a spinning wheel, the value of pi for that particular wheel changes to something less than 3.1415…

Everything in mathematics derives from definitions. Pi is what it is because we defined geometry in a particular way that made pi the value it is, it is not some intrinsic value of the universe. It doesn’t make it any less true (i.e. for any possible universe, the value of pi in Euclidian geometry is approximately 3.1415…, even if there was no universe at all), but we have to remember that there is nothing particularly special about the value we assigned to pi.

It’s true that “everything in mathematics derives from definitions”, but I think you have the definition of pi wrong–in geometry pi is defined as the ratio between diameter and circumference in Euclidean geometry, so the fact that this ratio may be different in non-Euclidean geometries does not actually mean that the value of pi can vary depending on the geometry, because pi is not defined as “the ratio of circumference to diameter in whatever space you happen to be using (or living in)”. What’s more, as you probably know pi shows up in other places besides geometry, so there are other ways of defining it; for example in complex analysis we have e^pi*i = -1, and in calculus there are various infinite series that sum to powers of pi, so for example a perfectly valid definition of pi is pi=4*(1 – 1/3 + 1/5 – 1/7 + 1/9 – 1/11 + 1/13 + …)

You do have some points, and hypnosifl has some counterpoints, but you can’t use special relativity for a spinning disk, because that’s not an inertial reference frame. (The points on the edge are subject to centripetal acceleration, and if it’s spinning at relativistic speeds, that acceleration of mv^2/r is far too much to ignore and approximate as inertial, so you need to go to general relativity.)

Special relativity does allow for non-inertial frames (like Rindler coordinates), but phenomena like length contraction and time dilation generally won’t follow the same rules in non-inertial frames as inertial ones (for example, you can’t assume that a clock with a greater coordinate velocity in a non-inertial frame is running slower, relative to the frame’s time coordinate, then one with a lower coordinate velocity). The boundary between special and general relativity is not defined in terms of what type of frame you use, but rather in terms of whether spacetime is curved or “flat”.

This is a re-imagining of an old debate that I believe was settled in the late 1800s.

Back then the question was phrased differently, but more to the point: is Math a psychological phenomenon? Gottlob Frege, the father of modern logic, basically said: that 1 + 1 = 2 is objectively true whether or not you recognize it as true… (More of his argument: http://plato.stanford.edu/entries/psychologism/#FreAntArg ). Modern logic came as a result, along with it, the challenge of uncovering the truth.

Continental philosophers like Badiou, hate the idea of any objective truth in the universe… even one as simple as 1 + 1 = 2 … as this has serious implications for any relativistic conceptions of truth and the foundation of many psychological arguments, and much, if not all of continental philosophy… while I think it’s blatantly true that the contents of math like numbers, equations, etc are a psychological phenomenon, what that language is referencing, or trying to reference are objective truths about the reality around us.

I imagine it’s true that most continental philosophers would deny mathematics has objective truth, but is that true of Badiou? I haven’t read his work but from summaries I thought he based his philosophy heavily on set theory–so even if his approach to deriving philosophical ideas from set theory is total nonsense, I thought that he at least took set theory to be some sort of fundamental foundation of reasoning. See the section of his wiki article on mathematics as ontology, as well as the diagram he drew for a lecture here which is basically incomprehensible without context, but does have the words “eternal truths” on the bottom border.

Badiou is quoted in the video as being against the idea of objective truth in mathematics. As far as I understand he dabbled in set theory to further his ontological conceptions… I don’t think he is a serious figure in set theory, and instead wrote away problems set theory created for him, ones that conflicted with his concept of being. Finally, the conception if ‘eternal truth’ is also constructed in badiou’s understanding of the world. Badiou doesn’t seem too interested in doing serious foundational work, and instead is using Marxism and lacanian psychoanalysis as much the foundation and inspiration of his ideas… Neither of which work in a world with objective truth.

You assert that the question was “settled” then offer zero justification for that assertion. How logical of you!

I could only take about five seconds of this, so perhaps he got around to addressing some of these points, but we might look to the etymology of ‘mathematics.’ Basically, it is the study of what we can learn, given a set of assumptions. The ‘reality’ of the results depends entirely on the assumptions, and the ‘reality’ that logic is a reliable method. It is another question entirely, why the universe seems so amenable to mathematical description, but the math would be equally real regardless.

In any event, it reminds me of an excellent paper: “How real are the real numbers”

I could only take about half a paragraph of this comment before I had to stop reading. :P

Even more wonderful is a little tome entitled “Mathematics Made Difficult” Carl E. Linderholm with wondrous chapters such as ‘So You Think You Know How To Count’. Sadly, I believe it is still out of print. You can still get a copy at Amazon, but it’s gonna cost ya.

Math is a language, and like any other lanuage exists. Rather than using words to describe the systems we observe, numbers and formulas are used to reprent the systems and their relationships.

If math is not real, and the internet is made of maths, I don’t know where the hell I am right now.

Math is real, if only because considering it a creation opens the door to IP concerns (which, arguably are there already, but let’s not make it easy on them!)

Too late! I’ve copyrighted the number ‘3’. I have sent takedown notices to the Three Little Pigs, as well as the US Federal government (with its 3 branches!) and the Catholic Church (Holy Trinity? That’s gonna cost ya. . . )

I’ve patented addition, subtraction, multiplication, and division. My rates will be reasonable, though.

At least Valve is safe from your predations for the forseeable future.

He doesn’t distinguish the 8 or 9 acceptions of the verb “to exist” and the 15 or 16 meanings of what is “real”.

Also, by talking so fast he tries to not allow the viewer to think what he has just said half a second before.

Some of the acceptions of “to exist”:

1. to have being or reality; to be
2. to eke out a living; stay alive; survive he could barely exist on such a low wage
3. to be living; live
4. to be present under specified conditions or in a specified place sharks exist in the Pacific
5. (Philosophy) Philosophya. to be actual rather than merely possible
b. to be a member of the domain of some theory, an element of some possible world, etc.
c. to have contingent being while free, responsible, and aware of one’s situation

The same guy could argue that since math is not really real, then imaginary numbers (by definition not real numbers) are not not real, therefore they are real, which makes math simultaneously not real and real.

No music? F this idea

Isn’t this just a riff on Gödel’s incompleteness theorem?

Not quite, I don’t think. But it’s close. The theorem basically states that, since, given that in any axiomatic system robust enough to admit simple arithmetic one will always be able to generate a true statement that is neither provable nor disprovable within that system and therefore your system is incomplete, meaning (basically) mathematicians will have something to do for all eternity :) I suppose that in that sense, you could say that math will always be there. Sorta.

I suppose that in that sense, you could say that math will always be there. Sorta.

With emphasis on the sorta.

Might one say that mathematics is an insoluble problem not seeking a solution? (It has done its due diligence and has consequently stopped looking)

Did I miss something?

I can’t be absolutely certain that mathematics has an objective existence outside of the models I build in my brain because I can’t be absolutely certain that ANYTHING has an objective existence outside of the models I build in my brain.

Cogito Ergo Sum, but everything else is provisional on some level.

That’s not to deny the possibility of the existence of an underlying objective reality, just to point out that I have no direct access to it.

I only have the models I build in my head from incoming sensory data, memory, imagination, and longing.

Now me, I use models that incorporate the notion of an inaccessible but nevertheless real “objective reality”, and I find science an excellent model-sorting technique in pursuit of models that more and more closely reveal its contours.

But I’m perfectly well aware that it’s ALL models inside my head. Because that’s all I’ve got.*

Is there any need, though, for a special discussion of this with reference to mathematics, particularly?

—-

* And that’s all you have, too, in the model I’m using. :-)

(*Just making the obligatory reference; I’m not attempting to criticise GlenBlank’s comments or anything.)

No it doesn’t exist, but it’s useful and sometimes even fun. The universe is irrational. It’s laws always changing.

I have always viewed math as formalized reason. It is as real as our own ideas are.

The universe operates according to rules and how those rules play out is sometimes computable, sometimes not, just as math is able to prove some things and not others.

Math begins where language ends. We have words for things and they must be learned by observation, or by processing combinations of other words.

The color red equates to our perception of a given frequency of light. Nobody knows whether or not we see the same red as beings, but language allows us to discuss that frequency of light with commonality to share thoughts.

Math goes one step beyond. Ever wish you could send people pictures, mind to mind?

That is what math does.

With language we can talk about nothing vs something and we can talk about the unit vs a few or many or some and even arrive at some commonality where a couple is seen by all who use the word in the same way, but a few becomes more difficult, many becomes more difficult still, leading to constructs like a little more than a few, but not so many that we would say many…

But those ideas are limited to the observable world and must be shared via some direct means. Without math, a couple needs to be demonstrated as in here is a unit and another unit and that is a couple units, but another unit still is a few units because it is not a couple…

Math takes us to a place where reason is formalized and with it we can establish some rules and apply them to communicate ideas with enough precision to automate them, reproduce them and come to understand the world in reproducible ways not dependent on rote learning from artifacts, pictures, stories and other indirect means and methods of sharing thoughts.

The closest thing we have in language to math is the analogy. We can say, this is like how that is and the implied dynamics carry from mind to mind, but again only when both minds have experienced and assigned language to common things. We do this with emotions and many social ideas that are difficult to express today with math, but I have no doubt there is a math for emotions should we ever decide to quantify them in some way to formalize them.

Ever have a discussion where the word hate comes up easily, but inside your mind you balk because hate has a “color” and “texture” and maybe nature that doesn’t quite fit the thing being discussed and you come to realize it’s actually loathing or disregard, both of which are ugly like hate is, but not primal and difficult to manage like hate is?

There is a math there waiting for us, should we formalize our emotional reason enough to make it possible.

When we grasp something well enough to quantify it, math is an artifact of that realization. We grok it then, and are able to take that understanding, formalize it, share with others and compute it, whatever it may be. The primary artifact of that is to express ideas more complex than direct observation will yield and with those infer something about the world not otherwise knowable or discoverable for simple lack of reason necessary to do so.

In this way we can compress our reason and use our limited intelligence to think about greater realizations or seek the aid of many or use machines to give names to complex ideas and share them and where those ideas prove resonant, name them and have the ability to communicate those as we do other basic language that requires shared experiences (both being there to agree on the name and what is experienced), or pictures or artifacts all of which cannot flow mind to mind like information can.

One day when we meet others and they have attributes different from ours we may learn things about ourselves and how we do things that are difficult to contemplate today. How would they have solved the problem of language and it’s limits it places on reason?

Would they need to invoke math, or can they communicate in ways we cannot?

Until then, math is as real as our thoughts are. I don’t see anything else as productive, though it is highly entertaining and stimulating to contemplate.

“Reality is that which, when you stop believing in it, doesn’t go away.” — Philip K. Dick

Math will not go away if you stop believing in it.

The numbering of years isn’t “wrong,” it’s simply not integer based. Really if we were using integers, not only would there be a year zero, but the subdivisions of the year: the months days and hours would be reversed. What we do now is analogous to describing negative fractions as -3 + 1/2 instead of saying -2.5

Okay, maybe the AD/BC system is not ‘wrong’ in so far as it did the job at the time, but it feels broken to me. Jesus would have had his first birthday in 2 AD.. The years 10 BC and 10 AD are 19 years apart. You would not do this with milestones. People might say they were in their ‘first mile’ when they started on a journey, like with year one.

For a long time, people shied away from the idea of zero. Even when the Arabic notation used a ‘zero’ (originally a dot) as a placeholder when you had hundreds and units but no tens, the zero number was not written by itself.
The French Revolutionary calender also started at year one. That was in 1792 AD. But I think the French knew their zero.

Maths is useful and helps us do stuff.

Pontificating about whether maths exists is not useful and does not help us do stuff.

Take your pick…

Wow this video is uninformed. I really can’t believe that someone paid for this video, and that anyone other than high school dropouts would repost it.

Math exists.

experience is the articulation of the world into various things – we understand things in terms of how they are different from each other – the contrast between different things is information – in that the way that something is different is what forms its identity – weather it be an object in a moment, or a pattern over time, matter in a space, or a person’s life.

math is the system that we use to process information – the idea that there could be one, two, or three etc is extrapolated only from things being different from each other. The basic concept that there are differences between things is necessarily true. if it exists only in our minds, then there is nothing that exists outside of our minds. if there was some objective reality outside of our minds in which there was no difference between things then our minds don’t have any relationship to that world.

Once you recognize that some people can process information quite nicely without any math at all, you’ll get more tail.

I better go have that beer now.

A famous statistician* once said that “All models are wrong, but some are useful.” I think something similar applies to mathematics.

There’s something profoundly naive about the idea that math is invented not discovered. Notation is invented. Math is discovered. Consider: I’ll build a bridge using what we currently call math. Now you invent a DIFFERENT math and use that to build a bridge. Either your math is a different way of stating all the same things as my math, or your bridge is not reliable, or else you built your bridge from experience and not from your invented new math.

Math is good and well but it’s not a patch on Maths.

The reason Europe has better education is because they buy their math in bulk.

I thought it was the Hylaean Theoric World?

Did you just perform a hypertrochian transquaestiation?

You two are giving me traumatic flashbacks.

The correct response being “Hey, not so fast!”

The slines aren’t going to get this.

I preferred the downwick version! Plato’s Ideal World of Forms thank you very much!

This is like required viewing for stoned college freshmen.

My guess is that he flunked out of philosophy and never got over it.

Gee, my guess is that he throughly enjoyed philosophy.

I once tried to construct the basic four operations using solely union and intersection. I haven’t trusted multiplication or division since.

For anyone with an interest in cognitive science, Lakoff and Nuñez’s

Where Mathematics Comes Fromis quite a read. Well worth your time, even if by the end you still find yourself disagreeing with them.Did you at least allow yourself recursive definitions and set membership?

{} = 0, { {} } = 1, { {{}}, {} } = 2, etc.

A + 0 = A

A + 1 = { A } ∪ A

A + (B + 1) = (A + 1) + B

That should be enough to get you addition and subtraction.

A * 0 = 0

A * 1 = A

A * (B + 1) = A * B + A

And there’s multiplication and division.

I’d like to say that I was just being excessively strict but really I was just incredibly naive. I didn’t even think to define the operand (in this case, the natural numbers).

This is not news. Aristotle wrote, in the earliest writing on mathematical philosophy, that numbers and the way we use them were nothing more than a social construction.

That said, if by “not real” you mean “social construction,” then pretty much almost everything humans have ever done is “not real.”

Plato wrote on mathematical philosophy before Aristotle, and of course he thought that mathematical truths were eternal, part of the realm of forms (the belief that math has some sort of objective reality is still known as “mathematical platonism”). And all human knowledge (including, say, “the Earth is round”) is of course a “social construction” in the broad sense that the process of humans coming to agreement about some fact necessarily involves social interactions, but hopefully you agree this doesn’t discredit the idea that there is such a thing as objective truth (in math or any other subject).

Why “hopefully”? While I think the idea of objective truth has a lot going for it I also think the idea that “objective truth” is self-contradictory has a lot going for it too. Why do we have to pretend to know the unknowable?

“Hopefully” because I think this would be a bad argument for showing that there can be no objective truth–I wasn’t expressing any hopes one way or another about whether you believe in objective truth or are agnostic/disbelieving, just commenting on this specific argument which some people might make.

“This is not news.”

Who said it was?

Also, the fact that this is not news

is not news.Not sure what hurt my head more…the ideas discussed or all that rapid fire editing!!

1. Now my brain hurts.

2. Is this guy related to Jason Lee in any way? Because his mannerisms and the rhythms of his speech really remind me of Jason Lee.

The unit exists. How many children do you have? One? Many? More than one? Not many? All set based concepts surrounding the concept of a single unit. Is the argument REALLY about how we convert the theory of units into a language? I don’t think it matters if we smell units, taste units, sing units, describe units as colors, or fart units. The undeniable fact that units exist and the study of units leads to an elegant language still remains. Just like grunting led to Shakespeare. However the details turned out, math was inevitable. Saying math may be a fictional creation is the same as saying sentience may be in our heads. What good does that kind of thinking do us.

There’s some condensed chunks of energy, a bunch of atoms, occupying a tiny bit of space in an incredibly small piece of their coincident progress along some entropic journey, commonly called ‘time’. And this is what you arbitrarily decide to call a child? Maybe those bits of stuff have better things to do.

“some…bunch…tiny bit…incredibly small…” – try to express our idea without any quantification or assumption of any form of unit.

There’s condensed chunks of energy, atoms, occupying space in a piece of their coincident progress along some entropic journey, commonly called ‘time’.

” chunk”. You can’t do it. We are slaves to the time stream.

If you take “chunks of” out of there, then, it’s devoid of unit or quantification. and still sensical.

I don’t think it’s an undeniable fact that units exist. In fact, this video put into language for me a world of thought I had been thinking, and having a hard time explaining to my friends for the last 15 years. I basically agree that if One exists (what you call The Unit), then all of Math exists. But I think that One-ness – like the circle – is an illusion that the human brain perceives because it is built to recognize patterns. (FYI, there are no circles in nature or reality. The human brain just forces you to perceive circle-like things as circles, and I posit that it does that same to things that appear unitary.)

You’re like, “Of course discrete units exist! Do I have one child, or do I have two?!?” And my reply is along the lines of the adage that “You can’t step into the same river twice.” Your one child is never, moment by moment, the same child that it was before its last big meal. We only perceive it as such because the human mind – mere software – breaks off these huge chunks of the universe and and forces things into the pattern of one-ness. Even at the strictly molecular level, atoms do not behave as discrete units – so as a predictive model, unitary mathematics do a shitty job of predicting reality.

But even though circles do not exist anywhere in nature or reality, we drive around in cars with round wheels that conform to our idea of what is “circular.” And so it is with One-ness or unitary-ness — our whole existence and experience depends on seeing it. An amoeba’s existence, not so much.

And we create equivalancy operations in which Mary and John are “equal” in a way that they can be “added” together to have two children.

Hmmm…you have me thinking about the time component of the unit. I have to agree that adds a twist. It’s very anthropomorphic of me to assume every sentient being co-exists with time in the same way we do. Really good point.

Regarding the circle. It’s just a variation on radial symmetry ( starfish) or the natural occurrence of the arc (Nautilus shell). Nature has MANY examples of the radial entity. Even on the molecular level atoms tend to arrange themselves in geometric patterns. So, I’m not so sure about that one.

“What good does that kind of thinking do us.”

Acknowledging that created concepts are not objective truths is a really, really important kind of thinking, and failing at that thinking has caused a hell of a lot of suffering and wrong-headedness throughout human history.

Also, ‘the unit’ exists as a social construction, which is why the term is as fluid as you describe. It doesn’t exist in any objective concrete way, it exists because I decide to delineate a unit. Which is why, say, my son could be a ‘unit,’ but from another subjective frame of reference, the family that contains my son could be a ‘unit.’ Or each cell that comprises my son could be a unit. etc.

I MIGHT have agreed with you up until the new tunneling quantum electron microscopes showed the atom in all its glory. The universe is indeed broken down into discrete units. The only way to say it isn’t is to say you exist outside of our time and space. At that point, we are waxing philosophic with no usable outcome for our efforts.

Oh, okay, Plato, so the atom is an indivisible monadic unit, now?

Posidonius, I think. Nope, Wikipedia says Leucippus (with mad props to Democritus the Mocker). But definitely not Plato!

I was just glibly dating his atomic model to Ancient Greece with an easily parsable name, rather than linking him explicitly to a specific thinker. Though Plato’s “simple bodies” aren’t terribly dissimilar to FrankenPC’s apparent understanding of the atom…

You are staring at the problem as you make your comments. There is perceived discreteness all around us. You’re just trying to cloud the issue with philosophical discourse. Have fun! Gained anything?

Acknowledging ambiguity isn’t clouding anything. Ignoring it, however, is.

You do know that atoms are largely empty space right? Quarks make up protons and neutrons, and quarks themselves are actually a quantum of energy. (wave-particle duality)

You might be able to argue that a quantum of energy is a discrete unit, but since it’s a wave, it’s sinusoidal, which is continuous. I.e., no discrete units!

Atoms are not discrete, they are made up of electrons and quarks. Electrons and quarks are not discrete; as best as physicists can figure out, they are probability distributions or at least act like them.

The unit exists, sure. Your perception is what breaks it up, like the way you can see the individual threads in a shirt if you have good vision, but it’s still a shirt. Maybe you perceive that you have many children, but to my eyes it’s still just your one family, and we’re both right from our own perspectives.

While the meat engine that makes up your own piece of the unit has the emergent property of a subjectively independent consciousness, it’s still part of the divine whole. I do believe I will go home and drink a beer in celebration of that (it’s important to have beliefs, you know). ^_^

Your perception breaks up the unit, sure, but it also forms it. a shirt is one shirt, a hundred threads, 50 threads of cotton with 50 threads of wool and some grams of colorants, a billion atoms and so on. who is to decide, objectively, which is the unit.

Besides our math is based on the biological limitations of our perception of reality. Its been speculated that being with more blurred sight (or senes) would have a less discret math.

Is math a truth about the universe or is math a fiction that comes out of our heads which is part of the universe so if the fiction of math is describing a consistency in out heads so then it’s a consistency about the universe and therefore a truth about the universe? I think I’m having a problem with the word “fiction”.

Would you agree that colour is a fiction? In that while there is a wavelength of light at 650 nm, the only thing that makes it ‘red’ is our perception of it. Nothing about that wavelength is intrinsically ‘red’.

How about the left- and right-wings of politics? Are they real or just fictional?

By the same token, even though mathematics seems to happen in the universe, we can’t break it down and point to the ‘mathematics particle’. It just seems that mathematics happens to be a model that exists beside the ‘real world’ and happens to describe it. At least, that’s the argument.

Personally, I tend to think that there’s something deeper happening and that mathematics is happening in some real way in reality. The reason for this is that purely mathematical applications of physics can predict things before we observe them – thus suggesting that they mathematics has been happening all along and we’ve just uncovered it.Of course, then there’s Godel’s Incompleteness Theorem – which the video touched upon when mentioning that there mathematical concepts that we are not capable of discovering. Godel’s Incompleteness Theorem states that there’s things which are mathematically true, but which can never be proven (no matter how complex the mathematical system used). (I’m massively simplifying because I’m not a mathematician.) Long story short – while mathematics seems to tell us things that are end up be true/real, it can’t tell us everything that’s true/real. Ever. No matter how good we get at it.

By the way for anyone that’s interested in this stuff, I think there are two books that are incredibly useful for informing further discussion:

1. Godel Escher Bach: An Eternal Golden Braid by Douglass Hofstadter. A non-fiction book with regular fictional interludes. Not necessarily the most accessible book, but everyone who I’ve ever met who’s read it has said it changed the way they think.

2. Anathem by Neale Stephenson. A fiction book with occasional non-fiction interludes. ‘Scififantasy’ has already made reference to it in this discussion, but a key component of the book is discussion of the concept of abstract ideas as being real things coming from somewhere.

I would urge anyone who’s had their mind blown by this stuff to read these books.

>Would you agree that colour is a fiction? In that while there is a wavelength of light at 650 nm, the only thing that makes it ‘red’ is our perception of it. Nothing about that wavelength is intrinsically ‘red’.

Perceiving color doesn’t seem anything like fiction. We perceive color fairly reliably to the point that we eventually were able to investigate it scientifically and recognize that the perception of red is a result of our eye seeing that wavelength of light. Fiction would at best be an analogy, and I don’t see how perception and fiction are truly analogous. Perception is the somewhat unreliable way sense organs discover sense objects, fiction is a deliberately false thing told to deceive, edify, or entertain.

The “red” part doesn’t exist anywhere but in our own heads as should be obvious if you think about color blindness for all of 20 seconds.

Existing in our head and fiction are two different things. Correct logical reasoning exists in our head but it isn’t fiction. There’s nothing obvious to me about the idea that perceptions are fictions.

You’re on the right track, now follow it! There are experiments you can do yourself that will prove to you that your perception of red is

independentof the wavelength of light… and it’s fun, too.Try this link and if you think it’s a video hoax, print out a copy of the Adelson Checkerboard and use a pair of scissors to prove it conclusively to yourself. What you perceive to be two different colors can often be

exactlythe same recieved wavelengths of light, or vice versa, and it’s a very complex cognitive process that creates the color in your mind.Color understood as qualia, is the subjective experience of sensory stimuli in consciousness conditioned by a lot of unconscious cognition (typically to speed processing up or filter less critical data). Objectively, we now have some understanding of how those qualia are formed based on eyes (or other sense organs) experiencing wavelengths with cognitive filters applied to sense data, and know that both the organs and the innate cognitive filters can be tricked at times. Illusions take advantage of these limits. That still doesn’t make qualia a fiction though, does it? It just means that senses can be unreliable.

Either “when a wavelength of X frequency strikes the eyeball of a normal healthy human s/he sees the color Y” is a true statement or it is not.

It is not true. This is easy to prove.

Therefore, any conclusions that rely axiomatically on color being determined by wavelength of light are

unproven. They are based on a false axiom. That’s the main point I was making.However, if you consider things that only exist in your head to be fictions, then all qualia must be fictitious. A bit different argument though innit?

There’s an organ in your brain the size of a kidney bean that determines your perception of color. The actual received wavelength of light has phenomenally little to do with this – if you spend four hours in a room with only green lighting, when you come out all your perception of color will have shifted. Read Oliver Sachs to learn more about how the idea that wavelengths = colors is fundamentally flawed. Colors are 100% created in the mind and it’s fairly easy to prove this, since they are perceived relative to the mind’s experiences and

notdirectly from what is presented to the eye at the time of perception.Second the recommendation of GED and Anathem, although I found the latter a little dry.

Very true.

Color is also dependent in the anatomy of the eye, depending on the amount and types of cones and canes in the retina.

If you wish to exclude colorblind people because they are not healthy or out of the (statistical) norm you still have to deal wit dogs that are unable to see the red spectrum of light, and zephyr falcons who can see ultra violet light.

“construct” is probably a better word than “fiction” (particularly since the word fiction typically implies a narrative).

But neither seems appropriate. I think nemomen had it right, it’s really perception. Both contruct and fiction seem to indicate “made-up” but really something that can be created in the mind’s eye can be interpreted many different ways. Perception is based on biology and (as long as your ocular instruments are working correctly) colors register the same.

Perception is based on biology and sociology acording to construtivist theory. In some east asian languajes blue and green are hues of the same color.

Math helps explain what something is and what something does but math is not the something itself.

That’s about as useful as saying “music tells us what notes to play and what effects they have, but music is not the notes.” Yes, it sort of is, if it’s to have any meaningful definition at all.

Is the word the thing? The word ‘tree’ is not the tree itself. So the musical notes written out isn’t the music but a description.

I think we’re all OK on the difference between description and referent; we’re discussing the nature of the thing itself, not the words we define with.

(I didn’t mean “musical notes written out”, or I would have said “score”. I meant “musical notes”, played as music. The music is a thing in itself, real even when not being played, which the score notates. Likewise, maths is a real thing in itself, an abstraction that corresponds to an actual property of the universe, even when it’s not being used to model physical objects.)

So you’re drawing a completely different distinction there – true, but not the point I was getting at.

The notes ARE NOT music though.

I run into so many Jazz geeks that know things, scales, names, transcribed solos, they have resonators inside their horns, they have experience and knowledge and can play many notes well BUT . . .

the notes is not the music.

We have multiple scientific, empirically-deduced physical models for the universe, which are nonetheless contradictory systems. There is the thing, and there is the model of the thing. Mathematics is a model, a semantic system for describing and organizing various systems, patterns, and concepts. A completely alien entity could, through observing and existing in the same universe, in theory come up with a completely different system for describing the same stuff (just as relativistic and quantum models of physics both describe the movement of bodies in the same universe, or how two different spoken languages both symbolically articulate sentient beings’ lived experience of the same universe).

A completely alien entity could also construct a completely different schema for describing, cataloging, and reproducing different frequencies of vibration travelling though gas. Their system would not necessarily be ‘music’ though it would be organizing the same set of phenomena that our ‘music’ does.

I have occasionally tried to speculate about beings whose existence is alien enough that what to us is simple arithmetic is to them complicated higher math and vice versa. Perhaps ones that exist as standing waves deep within a star.

Please, don’t give the Texas Schoolbook Commission any ideas.

I think the reality or fiction of mathematics as a whole doesn’t really make sense to talk about, because some concepts in mathematics are discovered realities, and some are invented fictions. You have to ask about individual concepts on a case by case basis.

For example, it seems like it is a discovered reality that the number of diameters of a circle it takes to measure out that circle’s circumference is about 3.14159 – the mathematical concept of pi. But other mathematical concepts, say the use of zero as a placeholder when writing decimal digits for example, didn’t come from any observation of nature, and you can’t really say they are “discovered”. They are invented techniques.

So it seems to me like you can’t just conflate all mathematical concepts together and say “it’s all natural reality” or “it’s all fiction”. Mathematics has both.

I think that’s a bit of a false dichotomy. Sure, the

notationis fiction, i.e. constructed for human convenience. But to point that out you compared apples and cars – the use of zero as a placeholder is a constructed convention, but theexistence of zeroisn’t, any more than the existence of pi.The set-theoretic definition of zero is constructed – but it was constructed for the purpose of formalising the observed properties of zero. I think any mathematician would say maths is discovered; only the

toolsof maths are invented.“…the use of zero as a placeholder is a constructed convention, but the existence of zero isn’t…”

Okay, you’ve just made an assertion. Back it up.

Fair enough; it’s certainly an assertion. I didn’t back it up because it felt like a fairly obvious one, just like the existence of “one”.

(Summary form only, because I can’t type much more today; somebody expand if need be.)

Cardinal numbers are innately accessible properties of the universe, as abstractions that apply to things in general. I can count three apples on my desk, and “threeness” is an actual property of the universe – cultures with no communication basis or common language can still come up with “three” independently, because they’ll both observe the same abstraction applying to different sets of three things.

If I throw away two apples, I’ll have “one” left, and that’s another observable abstract property. (We can’t ever

see“onenesss”, but we can observe it easily and frequently.)Eat that one, and I’ll have “zero” apples, just as I currently have “zero” orbital mind control satellites.

There are few human languages that don’t have word for this, because absence is a common and readily observable abstract property of objects. Once you’ve understood counting, absence, and subtraction, it’s not a great cognitive leap to realise that zero is a naturally-occurring number, just as three is. (Young children don’t automatically understand this, any more than they do quantity in general, but they learn it

veryquickly at the same time as they pick up other quantities.)The

notational systemthat uses zero as a placeholder is a clever human invention – but the use of zero corresponds to the universe in a deep way; the invention made properties of the universe easier to understand and record. (Note that post-zero base-n number systems are used because they’rer much more elegant than those that don’t – they express fundamental arithmetic and algebraic concepts cleanly where cruder systems modelled them much less well.)So: Our mathematical concept of “zero” is a notational and logical formalism, built by us as a tool, but it’s a

representationaltool, standing for an observable abstract property, in the same way that a stave stands for a piece of music, or “one” stands for an abstract set cardinality shared by the set of apples I’m currently eating.The existence of ‘one’ isn’t obvious though, that’s kind of the point of the whole video :)

His point is that you can, indeed view the existance of pi, or zero or even one as a fiction. The more math that you take, the more “abstract” it becomes. Eventually you circle back and realize that even “simple” math is an abstraction.

Oh, I understand his point, I just don’t think it’s true.

Yes, the more you understand maths the more you understand that the mathematical structure we use is an artificially created abstract logical formalism. (By the time you understand set theory, you can understand that “1+1=2” encodes a fantastically baroque set of chained formalism building from vastly simpler operations.)

But by the time you’re at that level, you should also understand that the structure is not

arbitrarilycreated, but has been designed and used precisely because it models “real-world” mathematical properties well.It gets trickier when we start to mess with the rules and model accuracy on purpose… and interesting how often that still turns out to be a good model for real properties.(Non-Eulidean geometry is the obvious non-mathematician’s example.)

As others have alluded to, like many conundrums of this sort, then answer to the question “does math ‘exist’.” depends on one’s definition of existence. You don’t have to fall down some “philosophers misinterpret quantum mechanics and conclude that our perceptions CREATE reality” rabbit hole to say that if NUMBERS don’t “exist,” than SPECIES don’t either. They’re both abstractions used to describe the objective world. But there’s certainly no obvious place to draw that line WITHIN mathematics. If the whole numbers are real properties of the universe, because we use them to count apples, than so are complex numbers when we use them to describe electromagnetic fields.

It’s maths. Maths, maths, maths maths, maths. With an S.

Was he indirectly riffing on Voltaire: If maths didn’t exist, we’d need to invent it.

And anthropomorphic deities.

Do you really expect the Empire to take language advice from one of our client states?

Why not mathematics over either choice? How does shortening it to maths or math make any kind of sense? How would you shorten other studies? Physics? Politics? Lingustics?

Physics: Physical studies.

Politics: Political studies.

Maths: Mathematical studies.

The pattern you present is inconsistent in its transformations. Adjusting for consistency, you get either:

Physics: Physical studies.

Politics: Political studies.

Mathematics: Mathematical studies.

-or-

Physes: Physical studies.

Pols: Political studies.

Maths: Mathematical studies.

Honestly, all you show is that ‘Mathematics’ is an acceptable shortening of “Mathematical Studies,” which does nothing to address the actual point of contention of whether “Math” or “Maths” is an adequate shortening of “Mathematics.”

I don’t actually agree with your assessment, but I’ll admit that I didn’t check any linguistic sources to provide those examples, and I agree they’re not really analogous.

Just thinking out loud.

“Otherwise, it sounds like you’ve got a lisp.”

Or it sounds like you come from a different country than tw1515tw, the narcissistic solipsist who thinks the entire world speaks exactly the same language s/he does on her/his small island nation.

Forget discussing whether Mathematics is a construct or an objective truth, we apparently need a video to explain to dullards that

spoken languageis a construct and not an objective truth.Do you also flip your shit when a Russian YouTube video says “математика” instead of “maths”? Did you really never learn that sometimes people from different parts of the world use different words, or different forms of similar roots, to describe the same object or concept?

Fucking math, how does that work?

This is not going to help me on my Chemistry test.

You should be studying … or at least working on a study calendar

OK, I’m sold. Click Subscribe.

For whatever definition of existence one chooses, I don’t really see how it’s productive to argue about whether math “exists” or not. All I know is that, given enough time, the probability of two ignorant but curious people to develop their own maths that are interchangeable seems to approach 1.

That is to say, math is essentially the same no matter who studies it and develops the mathematical systems. At least with the simple operations. 1+1 will always be 2, unless it’s 10, but base systems are still interchangeable anyway, so there’s still no difference. You can point to one thing, stick another thing next to it, and there will then be two things. And it happens every time. That sounds like to me like a discovery, not an invention.

Math is the best way to describe the universe around us. So unless there’s something better than math for understanding the world, I’m not convinced by any argument attempting to devalue its usefulness and our need for it.

Soooo… if an undisclosed amount of trees fall in the woods, and nobody is counting them, how much noise does it make?

It makes a barking lot of noise

“Does math exist in the universe or does it come from the human [mind]?” If you take the view that mind and matter are not actually different types, then math can be just spirit on another level, every bit as real as the universal one. This is objective idealism. I think that it makes the whole question drop away rather nicely!

Sort of… the problem is that it makes a lot of other interesting and useful questions go away as well.

As the mind consists of reasonable and evolutionarily necessary reversals of perspective away from the knowledge which the body and brain have of themselves and cosmology (ie the mind has a sense of meaning which is both complementary to and in opposition to that of the brain and body) in order to develop and refine tool making capacity maths is both real and unreal – like God. Really.

Pi seems to be inherent. I’m not a mathematician, but I can’t see how you could have any space with more than two dimensions without pi being the ratio of a circles circumference to it’s diameter. e is similarly constrained, and would show up anywhere you had rates of change that depended on one another.

Oh boy, do I have some bad news for you :)

There are plenty of “spaces” (in the formal mathematical sense) where pi has a totally different value than 3.1415…. The fact that it is that exact number is purely an artifact of us thinking that Euclidian geometry is the only “real” geometry.

Take, for instance, the so-called “Manhattan geometry”. Imagine a geometry where instead of being able to go anywhere, you could only go along the lines of a grid (like a car on the Manhattan street grid). Since a circle is defined as all the points at a certain distance (the radius) from the center of the circle, a circle in Manhattan geometry looks like this: http://en.wikipedia.org/wiki/File:TaxicabGeometryCircle.svg

I.e. a circle in Manhattan geometry looks like a diamond. For a radius r, the diameter is 2r and the length of the circumference is 8r, therefore the value of pi in Manhattan geometry is (exactly) 4.

But that’s not the real world, you say. The real world doesn’t use Manhattan geometry! In the real world, if you have a wheel, the length of the edge of the wheel divided by the diameter of the wheel will be (very close to) 3.1415….

No! Not necessarily true! Lets make use of one of the wonderful results of special relativity, which says that an object that is in motion contracts in length relative to a stationary observer (this is called “length contraction”, look it up on wikipedia if you want). That means that if the wheel is spinning very fast, from the perspective of the person standing still in the center, the moving outer edge is contracting in length, while the radius stays the same. In other words, for a spinning wheel, the value of pi for that particular wheel changes to something less than 3.1415…

Everything in mathematics derives from definitions. Pi is what it is because we defined geometry in a particular way that made pi the value it is, it is not some intrinsic value of the universe. It doesn’t make it any less true (i.e. for any possible universe, the value of pi in Euclidian geometry is approximately 3.1415…, even if there was no universe at all), but we have to remember that there is nothing particularly special about the value we assigned to pi.

It’s true that “everything in mathematics derives from definitions”, but I think you have the definition of pi wrong–in geometry pi is defined as the ratio between diameter and circumference

in Euclidean geometry, so the fact that this ratio may be different in non-Euclidean geometries does not actually mean that the value of pi can vary depending on the geometry, because pi isnotdefined as “the ratio of circumference to diameter in whatever space you happen to be using (or living in)”. What’s more, as you probably know pi shows up in other places besides geometry, so there are other ways of defining it; for example in complex analysis we have e^pi*i = -1, and in calculus there are various infinite series that sum to powers of pi, so for example a perfectly valid definition of pi is pi=4*(1 – 1/3 + 1/5 – 1/7 + 1/9 – 1/11 + 1/13 + …)You do have some points, and hypnosifl has some counterpoints, but you can’t use special relativity for a spinning disk, because that’s not an inertial reference frame. (The points on the edge are subject to centripetal acceleration, and if it’s spinning at relativistic speeds, that acceleration of mv^2/r is far too much to ignore and approximate as inertial, so you need to go to general relativity.)

Special relativity does allow for non-inertial frames (like Rindler coordinates), but phenomena like length contraction and time dilation generally won’t follow the same rules in non-inertial frames as inertial ones (for example, you can’t assume that a clock with a greater coordinate velocity in a non-inertial frame is running slower, relative to the frame’s time coordinate, then one with a lower coordinate velocity). The boundary between special and general relativity is not defined in terms of what type of frame you use, but rather in terms of whether spacetime is curved or “flat”.

This is a re-imagining of an old debate that I believe was settled in the late 1800s.

Back then the question was phrased differently, but more to the point: is Math a psychological phenomenon? Gottlob Frege, the father of modern logic, basically said: that 1 + 1 = 2 is objectively true whether or not you recognize it as true… (More of his argument: http://plato.stanford.edu/entries/psychologism/#FreAntArg ). Modern logic came as a result, along with it, the challenge of uncovering the truth.

Continental philosophers like Badiou, hate the idea of any objective truth in the universe… even one as simple as 1 + 1 = 2 … as this has serious implications for any relativistic conceptions of truth and the foundation of many psychological arguments, and much, if not all of continental philosophy… while I think it’s blatantly true that the contents of math like numbers, equations, etc are a psychological phenomenon, what that language is referencing, or trying to reference are objective truths about the reality around us.

I imagine it’s true that most continental philosophers would deny mathematics has objective truth, but is that true of Badiou? I haven’t read his work but from summaries I thought he based his philosophy heavily on set theory–so even if his approach to deriving philosophical ideas from set theory is total nonsense, I thought that he at least took set theory to be some sort of fundamental foundation of reasoning. See the section of his wiki article on mathematics as ontology, as well as the diagram he drew for a lecture here which is basically incomprehensible without context, but does have the words “eternal truths” on the bottom border.

Badiou is quoted in the video as being against the idea of objective truth in mathematics. As far as I understand he dabbled in set theory to further his ontological conceptions… I don’t think he is a serious figure in set theory, and instead wrote away problems set theory created for him, ones that conflicted with his concept of being. Finally, the conception if ‘eternal truth’ is also constructed in badiou’s understanding of the world. Badiou doesn’t seem too interested in doing serious foundational work, and instead is using Marxism and lacanian psychoanalysis as much the foundation and inspiration of his ideas… Neither of which work in a world with objective truth.

You assert that the question was “settled” then offer zero justification for that assertion. How logical of you!

I could only take about five seconds of this, so perhaps he got around to addressing some of these points, but we might look to the etymology of ‘mathematics.’ Basically, it is the study of what we can learn, given a set of assumptions. The ‘reality’ of the results depends entirely on the assumptions, and the ‘reality’ that logic is a reliable method. It is another question entirely, why the universe seems so amenable to mathematical description, but the math would be equally real regardless.

In any event, it reminds me of an excellent paper: “How real are the real numbers”

http://www.cs.auckland.ac.nz/~chaitin/olympia.pdf

I could only take about half a paragraph of this comment before I had to stop reading. :P

Even more wonderful is a little tome entitled “Mathematics Made Difficult” Carl E. Linderholm with wondrous chapters such as ‘So You Think You Know How To Count’. Sadly, I believe it is still out of print. You can still get a copy at Amazon, but it’s gonna cost ya.

Math is a language, and like any other lanuage exists. Rather than using words to describe the systems we observe, numbers and formulas are used to reprent the systems and their relationships.

If math is not real, and the internet is made of maths, I don’t know where the hell I am right now.

Math is real, if only because considering it a creation opens the door to IP concerns (which, arguably are there already, but let’s not make it easy on them!)

Too late! I’ve copyrighted the number ‘3’. I have sent takedown notices to the Three Little Pigs, as well as the US Federal government (with its 3 branches!) and the Catholic Church (Holy Trinity? That’s gonna cost ya. . . )

I’ve patented addition, subtraction, multiplication, and division. My rates will be reasonable, though.

At least Valve is safe from your predations for the forseeable future.

He doesn’t distinguish the 8 or 9 acceptions of the verb “to exist” and the 15 or 16 meanings of what is “real”.

Also, by talking so fast he tries to not allow the viewer to think what he has just said half a second before.

Some of the acceptions of “to exist”:

1. to have being or reality; to be

2. to eke out a living; stay alive; survive he could barely exist on such a low wage

3. to be living; live

4. to be present under specified conditions or in a specified place sharks exist in the Pacific

5. (Philosophy) Philosophya. to be actual rather than merely possible

b. to be a member of the domain of some theory, an element of some possible world, etc.

c. to have contingent being while free, responsible, and aware of one’s situation

The same guy could argue that since math is not really real, then imaginary numbers (by definition not real numbers) are not not real, therefore they are real, which makes math simultaneously not real and real.

No music? F this idea

Isn’t this just a riff on Gödel’s incompleteness theorem?

Not quite, I don’t think. But it’s close. The theorem basically states that, since, given that in any axiomatic system robust enough to admit simple arithmetic one will always be able to generate a true statement that is neither provable nor disprovable within that system and therefore your system is incomplete, meaning (basically) mathematicians will have something to do for all eternity :) I suppose that in that sense, you could say that math will always be there. Sorta.

With emphasis on the

sorta.Might one say that mathematics is an insoluble problem not seeking a solution? (It has done its due diligence and has consequently stopped looking)

Did I miss something?

I can’t be absolutely certain that mathematics has an objective existence outside of the models I build in my brain because I can’t be absolutely certain that ANYTHING has an objective existence outside of the models I build in my brain.

Cogito Ergo Sum, but everything else is provisional on some level.

That’s not to deny the possibility of the

existenceof an underlying objective reality, just to point out that I have no direct access to it.I only have the models I build in my head from incoming sensory data, memory, imagination, and longing.

Now me, I

usemodels that incorporate the notion of an inaccessible but nevertheless real “objective reality”, and I find science an excellent model-sorting technique in pursuit of models that more and more closely reveal its contours.But I’m perfectly well aware that it’s ALL models inside my head. Because that’s all I’ve got.*

Is there any need, though, for a special discussion of this with reference to mathematics, particularly?

—-

* And that’s all you have, too, in the model I’m using. :-)

something something xkcd something*

(*Just making the obligatory reference; I’m not attempting to criticise GlenBlank’s comments or anything.)

No it doesn’t exist, but it’s useful and sometimes even fun. The universe is irrational. It’s laws always changing.

I have always viewed math as formalized reason. It is as real as our own ideas are.

The universe operates according to rules and how those rules play out is sometimes computable, sometimes not, just as math is able to prove some things and not others.

Math begins where language ends. We have words for things and they must be learned by observation, or by processing combinations of other words.

The color red equates to our perception of a given frequency of light. Nobody knows whether or not we see the same red as beings, but language allows us to discuss that frequency of light with commonality to share thoughts.

Math goes one step beyond. Ever wish you could send people pictures, mind to mind?

That is what math does.

With language we can talk about nothing vs something and we can talk about the unit vs a few or many or some and even arrive at some commonality where a couple is seen by all who use the word in the same way, but a few becomes more difficult, many becomes more difficult still, leading to constructs like a little more than a few, but not so many that we would say many…

But those ideas are limited to the observable world and must be shared via some direct means. Without math, a couple needs to be demonstrated as in here is a unit and another unit and that is a couple units, but another unit still is a few units because it is not a couple…

Math takes us to a place where reason is formalized and with it we can establish some rules and apply them to communicate ideas with enough precision to automate them, reproduce them and come to understand the world in reproducible ways not dependent on rote learning from artifacts, pictures, stories and other indirect means and methods of sharing thoughts.

The closest thing we have in language to math is the analogy. We can say, this is like how that is and the implied dynamics carry from mind to mind, but again only when both minds have experienced and assigned language to common things. We do this with emotions and many social ideas that are difficult to express today with math, but I have no doubt there is a math for emotions should we ever decide to quantify them in some way to formalize them.

Ever have a discussion where the word hate comes up easily, but inside your mind you balk because hate has a “color” and “texture” and maybe nature that doesn’t quite fit the thing being discussed and you come to realize it’s actually loathing or disregard, both of which are ugly like hate is, but not primal and difficult to manage like hate is?

There is a math there waiting for us, should we formalize our emotional reason enough to make it possible.

When we grasp something well enough to quantify it, math is an artifact of that realization. We grok it then, and are able to take that understanding, formalize it, share with others and compute it, whatever it may be. The primary artifact of that is to express ideas more complex than direct observation will yield and with those infer something about the world not otherwise knowable or discoverable for simple lack of reason necessary to do so.

In this way we can compress our reason and use our limited intelligence to think about greater realizations or seek the aid of many or use machines to give names to complex ideas and share them and where those ideas prove resonant, name them and have the ability to communicate those as we do other basic language that requires shared experiences (both being there to agree on the name and what is experienced), or pictures or artifacts all of which cannot flow mind to mind like information can.

One day when we meet others and they have attributes different from ours we may learn things about ourselves and how we do things that are difficult to contemplate today. How would they have solved the problem of language and it’s limits it places on reason?

Would they need to invoke math, or can they communicate in ways we cannot?

Until then, math is as real as our thoughts are. I don’t see anything else as productive, though it is highly entertaining and stimulating to contemplate.

“Reality is that which, when you stop believing in it, doesn’t go away.” — Philip K. Dick

Math will not go away if you stop believing in it.

The numbering of years isn’t “wrong,” it’s simply not integer based. Really if we were using integers, not only would there be a year zero, but the subdivisions of the year: the months days and hours would be reversed. What we do now is analogous to describing negative fractions as -3 + 1/2 instead of saying -2.5

Okay, maybe the AD/BC system is not ‘wrong’ in so far as it did the job at the time, but it feels broken to me. Jesus would have had his first birthday in 2 AD.. The years 10 BC and 10 AD are 19 years apart. You would not do this with milestones. People might say they were in their ‘first mile’ when they started on a journey, like with year one.

For a long time, people shied away from the idea of zero. Even when the Arabic notation used a ‘zero’ (originally a dot) as a placeholder when you had hundreds and units but no tens, the zero number was not written by itself.

The French Revolutionary calender also started at year one. That was in 1792 AD. But I think the French knew their zero.

Maths is useful and helps us do stuff.

Pontificating about whether maths exists is not useful and does not help us do stuff.

Take your pick…

Wow this video is uninformed. I really can’t believe that someone paid for this video, and that anyone other than high school dropouts would repost it.

Math exists.

experience is the articulation of the world into various things – we understand things in terms of how they are different from each other – the contrast between different things is information – in that the way that something is different is what forms its identity – weather it be an object in a moment, or a pattern over time, matter in a space, or a person’s life.

math is the system that we use to process information – the idea that there could be one, two, or three etc is extrapolated only from things being different from each other. The basic concept that there are differences between things is necessarily true. if it exists only in our minds, then there is nothing that exists outside of our minds. if there was some objective reality outside of our minds in which there was no difference between things then our minds don’t have any relationship to that world.

Once you recognize that some people can process information quite nicely without any math at all, you’ll get more tail.

I better go have that beer now.

A famous statistician* once said that “All models are wrong, but some are useful.” I think something similar applies to mathematics.

*George E. P. Box

I’m just going to leave this here: Item #154 from the legendary HAKMEM.

“The universe is a twos-compliment machine….”

Indeed! Thanks for sharing.

There’s something profoundly naive about the idea that math is invented not discovered. Notation is invented. Math is discovered. Consider: I’ll build a bridge using what we currently call math. Now you invent a DIFFERENT math and use that to build a bridge. Either your math is a different way of stating all the same things as my math, or your bridge is not reliable, or else you built your bridge from experience and not from your invented new math.

tinyurl.com/l3cselt