Mathematics, Discovered or Invented?

[sayak83]

I do not think writing stuff in English will do the trick. Sure you can describe by words what the math symbols are doing. But it is still referring to the same mathematical operations only. For example what would be the non-mathematical way to describe the eigenvalue problem of quantum wavefunction relationships from which Energy or Momentum states are derived for a physical system?
As I see it, if it's a fictional game, it seems to be a necessary fiction to understand reality....which does not look like a property of fictional entities (like Superman of Santa Clause) or games (like chess or football).

Let me try another way
I would say something is real if it is mind independent. So, if suddenly all of us lose our memories and learning and concepts the things that continue to exist and happen, are real. If we all lose our memories, chess board and chess pieces become meaningless. So chess is a created fiction.
But the conservation of mass energy at the core of the Sun will still continue to follow E = mc2 relation and the Sun will continue to release photons and lose rest mass in accordance to this...whether we have forgotten about it or not. So that relationship is real. The relationship has the physical variables (mass, Energy, Velocity) and the mathematical operations relating them. One cannot say that the physical variables only is real and the math relational operation itself is a fiction....can one. Both have to be mind independent if the physics has to hold irrespective of whether minds can recall it.
However for the mathematical relations built into this energy equation to be mind independent the corresponding mathematical system on which their truth values rest have to be mind independent. It cannot be the only specific math cases are mind independent and the rest which are not instantiated in a science equation are not. The ontology of the entire structure or game has to be one and the same.
This, to me, suggests that the mathematical systems are mind independent systems. The only other alternative is that these physical relationships and physical entities present in the scientific theories are all themselves mind dependent fictions and energy, mass etc become meaningless as soon as minds forget about them.

 

[vulcanlogician]

Let me try another way

Excellent post. And good questions! I envy people who can think as precisely as you do.

@Polymath257

I hope this isn't becoming a "let's bother Polymath with all our questions" thread. But I do have one more objection if you don't mind.

What about fractals? Fractals were studied at first for their mathematical beauty, but, as it turns out, fractal geometry features in the natural world. When scholars first began to understand fractals, they had no idea that they could be descriptive of nature. But as it turns out, they are. Plant life in particular conforms to the patterns of fractal geometry. But fractals weren't "discovered" by examining plant life. And it's not just plant life that fractals describe. Many features of the natural world adhere to the patterns of fractal geometry.

I'd like to say that this suggests that there is a direct link between the natural world and mathematics. Simply because we human understand this and express this (because it's the truth) doesn't mean that fractal geometry is a sort of "language" whose purpose is to express fractal things because humans wish to discuss fractals. It suggests that fractals are a mathematical discovery that also happen to accurately describe certain parts of the natural world.

I'm CERTAIN that there is a more salient example where the pursuit of mathematical beauty has led to conclusions that accurately describe the natural world, but I can't think of it right now. Some guy, exploring beautiful mathematical structures... then boom. They turn out to be a feature of the natural world.

But doesn't that sort of challenge both your "language" and "game" conjectures? After all, you can study language all you want, but studying a language won't wind up (accidentally) furnishing you with the truth about the planet Jupiter or any other part of the natural world. Studying language tells us mostly about what we humans want to communicate and how we communicate those things. You won't learn anything about Jupiter's physical properties by reading Shakespeare.

Not so with mathematics. You could be studying some abstract beautiful concept that (as far as you know) only exists on paper in mathematics. But lo and behold! As physicists advance their understanding of the universe, they may sometimes find that your "on paper mathematics" ACTUALLY applies to the physical universe. Hmmmmmmm.....

If you want to treat your game analogy, Let's suppose you were a hermit who lived your life in a cottage. You were never interested in setting foot in the outside world. So, being the hermit you are... you got bored one day and invented the game of chess. After having invented the game, you (uncharacteristically) left your cottage to find someone to play with. (Because chess is fun, and you need two players.) But when you stepped out of the cottage door, you find that the outside word is an 8 by 8 grid where two factions are warring with each other. Each faction consists of a King, queen, two bishops, two knights, and two rooks (and 8 other dudes who kinda don't matter in the long run, but are extremely useful sometimes). Isn't that a little too much similarity to be considered coincidence? It's AWFULLY coincidental that mathematics, if it is to be deemed some "invention" does SUCH a good job at describing what the world really is.

I'm not talking about when physicists use mathematics to determine things about nature, even though, that fact in itself is compelling. I'm just saying, isn't it weird that a mathematician can play around with shapes, lines, and structures at home in their notebooks, and then-- OOPS! Turns out they ended up describing the natural world. Seems a little to coincidental to me. And it makes me sympathize with sayak83's point.

I don't mind mind if you tear my objections to shreds. Shred away! But I still think these are reasonable objections to mathematical fictionalism.

 

[vulcanlogician]

Euclid had 5 postulates for geometry.

And for those non-nerds out there, I'd like to point out Polymath's use of the word "postulate" in response to my claims about "axioms."

An axiom is a self evident thing. Something that is obviously true. "Self-evident" is a term that gets tossed around a lot.

A postulate is similar to an axiom. It serves the same purpose (more or less) in a logical expression. But a postulate is something that "we are going to assume to be true for the sake of an argument." -- That's different from something that is undeniable and obvious.

Are there any axioms in your worldview, Poly? Or is it postulates all the way down?

 

[Twilight Hue]

One of the age-old questions that we haven't debated recently (at least not in the last year).

I think that mathematics is discovered. One piece of evidence is that maths has been discovered multiple times independently.
We assume it even to be so universal that aliens on other planets must have discovered it, if they have technology.

Tagging @Polymath257

Pythagoras is said to be the world's first mathematician.

 

[Heyo]

Pythagoras is said to be the world's first mathematician.

Pythagoras was very late to the game, though it can be said that he first formalised geometry and arithmetic. Thus making him the first to do maths insteadof simply calculating.

 

[Twilight Hue]

Pythagoras was very late to the game, though it can be said that he first formalised geometry and arithmetic. Thus making him the first to do maths insteadof simply calculating.

Makes you wonder where his knowledge originated.

It's a shame much of history gets lost over time.

 

[Heyo]

I think it might be relevant to say what is considered to be 'basic math'.

Yep, I said so in my first post to you.

Another take on this is that some ideas (one, zero, simple addition) are ways that humans think. That does NOT mean it is 'discovery', but rather it is a form of species bias.

Jung's archetypes? Then how come that AIs discover the same laws and relations?

I would disagree. Even you noted that calculus would be part of any basic system.

I would put calculus at the point of investigation. I think anything more basic than calculus is discovered. From that point on, I'm willing to concede invention.

And yet, more modern investigations show that it is NOT the only way to do math. Even logic can be done in multiple ways (and has been in human culture).

Those modern maths and logics are invented for the purpose of investigation, they don't "come natural".

 

[vulcanlogician]

Pythagoras was very late to the game, though it can be said that he first formalised geometry and arithmetic. Thus making him the first to do maths insteadof simply calculating.

Very true the Egyptians and Sumerians had understood many geometric concepts centuries before the Greeks. And they applied these "truths" in order to help them more efficiently perform commerce and agriculture. But the Greeks had a peculiarity. A peculiarity that still remains with us inj modern science. The Greeks sought to understand these concepts via a theory... as a universal... not just some thing that led to a better crop yield. But a universal truth that applied where it was relevant-- whenever it was relevant.

Pythagoras, Euclid, and Plato must be understood post this Greek conceit in order to really appreciate their value. Not that the Sumerians and Egyptians didn't offer valuable contributions. They just failed to understand that these mathematical facts were universal not just practical.

 

[Polymath257]

And for those non-nerds out there, I'd like to point out Polymath's use of the word "postulate" in response to my claims about "axioms."

An axiom is a self evident thing. Something that is obviously true. "Self-evident" is a term that gets tossed around a lot.

A postulate is similar to an axiom. It serves the same purpose (more or less) in a logical expression. But a postulate is something that "we are going to assume to be true for the sake of an argument." -- That's different from something that is undeniable and obvious.

Are there any axioms in your worldview, Poly? Or is it postulates all the way down?

In math today there are only postulates (according to this terminology). We call them axioms, but no longer see them as self evident truths.

 

[wellwisher]

Didn't too well at maths then?

I was part of an Honorary Math Society in High School based on US National Testing; done in the Junior Year before college applications. However, my favorite teacher was in Chemistry, so I liked Chemistry better in High School. Chemistry was similar to math and language with the 94 natural elements, like the letter of a natural alphabet, that fit together into various words; molecules. Each letter of the chemical alphabet was fancy with lots of dynamics sub-aspects.

Oxygen is the most common vowel in this element alphabet, since it reacts with almost everything, with H2O or a blend of oxygen and hydrogen the foundation of the Chemistry. All reactions are called oxidations or reductions; based on O or H, even if O and H are not involved, since they set the pattern for all the rest. Modeling life on the foundation of Chemistry; water, makes a lot of sense; shoulders of giants. Water is the most researched substance in all of science. It has the most extensive tech support; tools.

I also enjoyed applied science in college, so I became a Development Chemical Engineer. What I found was that I could work faster without math since math although useful to help haul things, math does not invent, beyond what it was designed by someone to haul. My math horse was more for having fun; satisfy the system.

I look at Chromosomes of DNA, immersed in water, as a paragraph of atoms mingling and sitting on the foundation of chemistry. Water has more than the needed specs, to carry the DNA load. DNA is about template perfection and is not mostly about dice and cards; gaming math psychosis. Water is stable and unchanging, therefore it instills that upon the DNA; pushed to be unchanging.

Much of math is connected to data plotting. All data plots are math equations ready to be applied or discovered. You run experiments, plot the data and then apply or develop equations to express the data curve. I useful data plot for many things was the Log. If the math is good, this can be useful to others, who do similar experiments or who wish to build upon previous experiments.

Where calculus comes is about analyzing these data curves. Differentiation is the slope on the curve at any given point, while Integration is the area under the curve from A to B. As curves get more complex so do the rules for the calculus of curve analysis. A sharp slope change can reflect a sudden change at the molecular level. That exact slope angle can be useful.

From this foundation you can do engineering applications. We can derive the ratio of volume to surface area for simple shapes or even more complex shapes. Say we dug a long tunnel and used parabolic archways, we can integrate the equation of the parabola from A to B to get the area under the curve, to make sure we have enough air flow in the tunnel. Air flow is where ChemE math comes in; fluid dynamics. Is the air flow gentle; lamina, or turbulent? Turbulent is better for air mixing so add some fans.

The most advance math I had to do was called Continuum Mechanics. This is essentially calculus is 3-D, so you can analyze the properties of materials and thermodynamics that are changing in continuous flow chemical reactors. If you react A and B to get C, all your parameters change in 3-D, as a function of distance along the reactor, as well as radially, from the walls of the reactor; thermodynamics, mass and heat transfer, fluid dynamic properties, reaction kinetics, etc. Process control may have many zones along the length to compensate. The goal is a good quality product at maximum production, by controlling pressure, flow rates of reactants, removal of by-products, adjustments of the temperature profile, better mixing, etc.

If you add to this the creation of nonNewtonian fluids; plastics, one may need to do other things, such as add a screw in the pipe to assist the movement of the thicker material without adding pressure at the beginning that can alter the polymerization kinetics; over heat. Moat of this is making things that are not natural to the earth, and therefore they require math that was extrapolated from natural things, to simulate unnatural things. If math was natural it would not work for the unnatural, just as well as for the natural. We need to lead that horse of math with different assumptions. Often this results in unique proprietary math horses that give a market edge.

 

[Mock Turtle]

I was part of an Honorary Math Society in High School based on US National Testing; done in the Junior Year before college applications. However, my favorite teacher was in Chemistry, so I liked Chemistry better in High School. Chemistry was similar to math and language with the 94 natural elements, like the letter of a natural alphabet, that fit together into various words; molecules. Each letter of the chemical alphabet was fancy with lots of dynamics sub-aspects.

Oxygen is the most common vowel in this element alphabet, since it reacts with almost everything, with H2O or a blend of oxygen and hydrogen the foundation of the Chemistry. All reactions are called oxidations or reductions; based on O or H, even if O and H are not involved, since they set the pattern for all the rest. Modeling life on the foundation of Chemistry; water, makes a lot of sense; shoulders of giants. Water is the most researched substance in all of science. It has the most extensive tech support; tools.

I also enjoyed applied science in college, so I became a Development Chemical Engineer. What I found was that I could work faster without math since math although useful to help haul things, math does not invent, beyond what it was designed by someone to haul. My math horse was more for having fun; satisfy the system.

I look at Chromosomes of DNA, immersed in water, as a paragraph of atoms mingling and sitting on the foundation of chemistry. Water has more than the needed specs, to carry the DNA load. DNA is about template perfection and is not mostly about dice and cards; gaming math psychosis. Water is stable and unchanging, therefore it instills that upon the DNA; pushed to be unchanging.

Much of math is connected to data plotting. All data plots are math equations ready to be applied or discovered. You run experiments, plot the data and then apply or develop equations to express the data curve. I useful data plot for many things was the Log. If the math is good, this can be useful to others, who do similar experiments or who wish to build upon previous experiments.

Where calculus comes is about analyzing these data curves. Differentiation is the slope on the curve at any given point, while Integration is the area under the curve from A to B. As curves get more complex so do the rules for the calculus of curve analysis. A sharp slope change can reflect a sudden change at the molecular level. That exact slope angle can be useful.

From this foundation you can do engineering applications. We can derive the ratio of volume to surface area for simple shapes or even more complex shapes. Say we dug a long tunnel and used parabolic archways, we can integrate the equation of the parabola from A to B to get the area under the curve, to make sure we have enough air flow in the tunnel. Air flow is where ChemE math comes in; fluid dynamics. Is the air flow gentle; lamina, or turbulent? Turbulent is better for air mixing so add some fans.

The most advance math I had to do was called Continuum Mechanics. This is essentially calculus is 3-D, so you can analyze the properties of materials and thermodynamics that are changing in continuous flow chemical reactors. If you react A and B to get C, all your parameters change in 3-D, as a function of distance along the reactor, as well as radially, from the walls of the reactor; thermodynamics, mass and heat transfer, fluid dynamic properties, reaction kinetics, etc. Process control may have many zones along the length to compensate. The goal is a good quality product at maximum production, by controlling pressure, flow rates of reactants, removal of by-products, adjustments of the temperature profile, better mixing, etc.

If you add to this the creation of nonNewtonian fluids; plastics, one may need to do other things, such as add a screw in the pipe to assist the movement of the thicker material without adding pressure at the beginning that can alter the polymerization kinetics; over heat. Moat of this is making things that are not natural to the earth, and therefore they require math that was extrapolated from natural things, to simulate unnatural things. If math was natural it would not work for the unnatural, just as well as for the natural. We need to lead that horse of math with different assumptions. Often this results in unique proprietary math horses that give a market edge.

If this is all true it is just so surprising as to why you seem to hold some of your views then - apart from a religious belief interfering with your thought processes perhaps.

 

[Polymath257]

Makes you wonder where his knowledge originated.

It's a shame much of history gets lost over time.

Pythagorus is said to have traveled extensively, so had access to fairly extensive Egyptian and Babylonian records.

 

[Polymath257]

Yep, I said so in my first post to you.

Jung's archetypes? Then how come that AIs discover the same laws and relations?

I would put calculus at the point of investigation. I think anything more basic than calculus is discovered. From that point on, I'm willing to concede invention.

Those modern maths and logics are invented for the purpose of investigation, they don't "come natural".

I find it interesting where people will draw the line between discovered math and invented math. For you, it is calculus. For @sayak83 it seems to be partial differential equations (which are used extensively in quantum mechanics).

I see both as extensions of language that we use to model things we are interested in. But I have very serious doubts about the 'existence' of these systems outside of our minds.

For example, you mention the natural numbers as 'God given' (following Kronecker). But take a look at Graham's number: Graham's number - Wikipedia

If we suppose that the observable universe is one trillionth of the 'real universe' and that ours in only one of 10^10^10^10 possible universes (far more than string theory postulates), and if we imagine each of these universes lasting 100 trillion years, and if we count the number of Plank units (plank volume times Plank time) for each of these universe, then the number of possible rearrangements of all of these Plank units throughout all of the possible universes is far, far, far smaller than Graham's number.

Given that, in what possible sense does Graham's number 'actually exist'?

As far as I can see, it *only* exists in our minds as a construct in a formal system. anything outside of that is excluded by its size.

My conclusion is that even the natural numbers cannot ALL exist. Only a very small part of that collection can 'really exist'.

The same can be said for the real numbers, differential equations, etc.

So, in answer to @vulcanlogician and @sayak83, I think we *invent* formal systems to mimic what we see. We are then amazed that those formal systems continue to mimic what we see. But those actual formal systems go far, far beyond what is testable. They have 'objects' that are simply too large to actually exist in any conception of a multiverse. But these systems are extensions of our ability to have language. We *invent* the ways to describe things. We choose systems that are flexible in their ability to describe. And that vast, vast majority of what those formal systems have has *nothing* to do with anything physical even in the remotest extension we can imagine.

So, no, I am not surprised that occasionally our patterns of thought show up in how we look at the universe around us.

 

[Polymath257]

If you want to treat your game analogy, Let's suppose you were a hermit who lived your life in a cottage. You were never interested in setting foot in the outside world. So, being the hermit you are... you got bored one day and invented the game of chess. After having invented the game, you (uncharacteristically) left your cottage to find someone to play with. (Because chess is fun, and you need two players.) But when you stepped out of the cottage door, you find that the outside word is an 8 by 8 grid where two factions are warring with each other. Each faction consists of a King, queen, two bishops, two knights, and two rooks (and 8 other dudes who kinda don't matter in the long run, but are extremely useful sometimes). Isn't that a little too much similarity to be considered coincidence? It's AWFULLY coincidental that mathematics, if it is to be deemed some "invention" does SUCH a good job at describing what the world really is.

I'm not talking about when physicists use mathematics to determine things about nature, even though, that fact in itself is compelling. I'm just saying, isn't it weird that a mathematician can play around with shapes, lines, and structures at home in their notebooks, and then-- OOPS! Turns out they ended up describing the natural world. Seems a little to coincidental to me. And it makes me sympathize with sayak83's point.

I don't mind mind if you tear my objections to shreds. Shred away! But I still think these are reasonable objections to mathematical fictionalism.

Let's change this analogy slightly. Suppose that someone watched how Kings and Queens and Knights managed war and found some basic rules. They then called this set of rules 'the math of war'. Suppose those rules turn out to be the rules we use for the game of chess.

Should someone of that culture then be surprised that a particular game of chess matches the moves of some particular war?

Suppose that not every war was described that way, but some additional rules were then found to be added on to those of chess, which extended the range of applicability of those rules. Should someone then be surprised that a new war followed those extended rules?

Suppose next that what they had previously used English to describe was more compactly described by those rules they invented to describe the original wars. Suppose those rules are very flexible and provide a host of different structures to mimic the way wars are done. Should we then be surprised that some extension of those rule was then able to describe some trade disputes?

Once you have a sufficiently rich language that can be managed in your formal rules, is it then surprising that *some* of the poetry that is done for its own sake is found to be applicable to real situations?

I take an example of Greek philosophy. So many different variants of possible ways to describe reality were attempted that it shouldn't be surprising that *one*, say the atomic theory, is somewhat similar to what was later found to be the case. Aristotle's physics is simply wrong, but that of Democritus has some aspects that remain. if you try 1000 different permutations of the possibilities, I don't find it surprising that one is found to be close to the reality.

 

[Heyo]

I find it interesting where people will draw the line between discovered math and invented math. For you, it is calculus. For @sayak83 it seems to be partial differential equations (which are used extensively in quantum mechanics).

As long as we don't insist on an extreme, the border is most probably arbitrary and fuzzy.

I see both as extensions of language that we use to model things we are interested in. But I have very serious doubts about the 'existence' of these systems outside of our minds.

I see the "things we are interested in" as existing independent of our minds - and the language we use to describe them as arbitrary - as long as we can translate the descriptions without gaps and error, they are functionally the same.

For example, you mention the natural numbers as 'God given' (following Kronecker). But take a look at Graham's number: Graham's number - Wikipedia

If we suppose that the observable universe is one trillionth of the 'real universe' and that ours in only one of 10^10^10^10 possible universes (far more than string theory postulates), and if we imagine each of these universes lasting 100 trillion years, and if we count the number of Plank units (plank volume times Plank time) for each of these universe, then the number of possible rearrangements of all of these Plank units throughout all of the possible universes is far, far, far smaller than Graham's number.

Given that, in what possible sense does Graham's number 'actually exist'?

As far as I can see, it *only* exists in our minds as a construct in a formal system. anything outside of that is excluded by its size.

My conclusion is that even the natural numbers cannot ALL exist. Only a very small part of that collection can 'really exist'.

They aren't "real", but they exist, and they exist in the realm of the Platonic ideals - which is not limitet the same way the real world is.

The same can be said for the real numbers, differential equations, etc.

So, in answer to @vulcanlogician and @sayak83, I think we *invent* formal systems to mimic what we see. We are then amazed that those formal systems continue to mimic what we see.

And the fact that they keep being consistent for everyone is a sign that there is something like objective existence to the systems.

But those actual formal systems go far, far beyond what is testable. They have 'objects' that are simply too large to actually exist in any conception of a multiverse. But these systems are extensions of our ability to have language. We *invent* the ways to describe things. We choose systems that are flexible in their ability to describe. And that vast, vast majority of what those formal systems have has *nothing* to do with anything physical even in the remotest extension we can imagine.

So, no, I am not surprised that occasionally our patterns of thought show up in how we look at the universe around us.

 

[Polymath257]

As long as we don't insist on an extreme, the border is most probably arbitrary and fuzzy.

I see the "things we are interested in" as existing independent of our minds - and the language we use to describe them as arbitrary - as long as we can translate the descriptions without gaps and error, they are functionally the same.

And I agree when it comes to physical things: they exist outside of our minds.

They aren't "real", but they exist, and they exist in the realm of the Platonic ideals - which is not limitet the same way the real world is.

Ok, that makes no sense to me. I only see “platonic ideals” as existing in our minds.

And the fact that they keep being consistent for everyone is a sign that there is something like objective existence to the systems.

I see it as more that we have fairly standardized ways of thinking.

The fact that we took 2000 years to find out that Euclidean geometry isn’t the only one possible is evidence of this.

 

[Heyo]

And I agree when it comes to physical things: they exist outside of our minds.

As do ideal things.

Ok, that makes no sense to me. I only see “platonic ideals” as existing in our minds.

They don't exist in reality, but there is a realm where necessarily existing things, that are not real, live - like forms and numbers. We don't see them with our eyes, but with our mind, that's why you might think they originate from there. But like real things, forms and numbers exist still, even if we don't "see" them at the time.

I see it as more that we have fairly standardized ways of thinking.

Who is "we", Earthlings? At least the concept of numbers exists in non-human animals, too.

The fact that we took 2000 years to find out that Euclidean geometry isn’t the only one possible is evidence of this.

Or it is evidence that Euclidean geometry is the "natural" way of shapes existing.

 

[wellwisher]

Let me try another way
I would say something is real if it is mind independent. So, if suddenly all of us lose our memories and learning and concepts the things that continue to exist and happen, are real. If we all lose our memories, chess board and chess pieces become meaningless. So chess is a created fiction.
But the conservation of mass energy at the core of the Sun will still continue to follow E = mc2 relation and the Sun will continue to release photons and lose rest mass in accordance to this...whether we have forgotten about it or not. So that relationship is real. The relationship has the physical variables (mass, Energy, Velocity) and the mathematical operations relating them. One cannot say that the physical variables only is real and the math relational operation itself is a fiction....can one. Both have to be mind independent if the physics has to hold irrespective of whether minds can recall it.
However for the mathematical relations built into this energy equation to be mind independent the corresponding mathematical system on which their truth values rest have to be mind independent. It cannot be the only specific math cases are mind independent and the rest which are not instantiated in a science equation are not. The ontology of the entire structure or game has to be one and the same.
This, to me, suggests that the mathematical systems are mind independent systems. The only other alternative is that these physical relationships and physical entities present in the scientific theories are all themselves mind dependent fictions and energy, mass etc become meaningless as soon as minds forget about them.

The relationships in nature are independent of our mind. They do not need humans to exist. The system we use to represent the relationships of nature and realty, which in this case is math, is more like a bridge between mind and matter. This bridge analogy makes math very similar to a type of language based on numbers, letters and symbols.

The bee flying from flower to flower is a real thing. If I wished to explain what I just saw to you, I may use verbal language and say. "I saw a bee fly from flower to flower." This audio output will create a mental picture so you can sort of see, without being there.

We use the math language to express this same physical reality, but in far more detail; "the bee is flying from flower to flower." The math language allows an even more accurate experience of reality, by drawing a curve of the flight path. Even if you are not present to see it, a mental image appears from the equation, that has a similar effect on your consciousness. Spoken language would have a hard time, explaining with words, the meandering flight of the bee, since even the word meandering may not induce the same visualization in all; zig zag, loop to loop, etc. But the math equation, can be plotted, to see the exact curve. Math extends spoken language.

Math is very similar to spoken language, but digs even deeper, into how natural and manmade things relate. Whereas the verbal language can be spoken, written and read, the math language is more or less restricted, to reading and writing. It is not used very much for verbal use. Few people talk in math equations, except math nerds and math majors. Math uses a different language bridge; read/write, which is more visual than verbal. This visual connection; read/write, explains why math was historically more male orientated; males are more visual; math, while females are more verbal; novels and words.

Verbal languages are subjective, as inferred by there being 6-7K human languages on the current earth. Any sound or any symbol can be used for anything. The sounds do not have to have a natural sound-thing-action-modifier connection. I never heard a cat say the sound "cat", so why that sound? It is subjective and arbitrary. If we all agree, it can still be useful.

Math by being more visual, is connected to the more natural universal language of vision. Our eyes can pick up a wide range of photons, at the visible spectrum wavelengths, in various geometries, texture, patterns, tones, gradients, etc. This natural language; that comes with the eyes, is used by math, thereby making math more universal; data plots. Math is universal in the sense of using a natural bridge to reality; natural visual language bridge, untainted by the subjectivity of the arbitrary sounds of the verbal languages. However, our understanding of the math language bridge is still based on our conceptual foundations of thought.

If you made a new equation, never seen before, you will need to set the stage, as to what others are seeing, before it makes any sense and it can then be applied. The driver needs a goal, before the math horse will start to walk.

The analogy would be a large wall sized mural that symbolizes all of science. The new equation is about a small piece on that wall. I can use spoken language to narrow your search to a given area. After that, spoken words may get in the way. However, I will need to shift gears and explain my premises for approaching that area of the mural and my data, that way. After that you can come to a focus with the stage is set. You can now explore more deeply, where words alone, cannot go.

A good example of the driver leading the horse of math; equations, is comparing Newtonian Gravity with Einstein's Gravity; General Relativity. Both math models describe the same phenomena on the mural of science; gravity. Yet the two sets of equations are different. This has to do with two different sets of foundation premises. Both correlate fine at the conditions common to humans and the human senses. General Relativity is more accurate, when we have extra tools so see extreme conditions not normal to earth and our five senses. With the two stages set, we can jump off Newtonian and enter the strange world of Relativity and see that you normally cannot see. Or we can jump back.



I often complain, "all the time", about statistical math and the life sciences. This is mostly due to too many people thinking math is natural reality and less a useful bridge, between mind and matter, that uses the universal visual language, between mind and matter. However, it is still dependent on our foundation premises. The psychological problem, I see is that if you assume "math is reality" and not just the language bridge between reality and mind, you will base life and consciousness, on the math premises of dice and cards. You will lose touch with natural reality. Life is about order. Life cannot coordinate trillions of cells, in the land of jackpots and casino play. Why approach life with based on the premises of known exceptions? That is illogical to the preponderance of the observational data.

What appears to have happened is the foundation premises, for the collective mental stage, is set by the premises of a randomizer math bridge. This is math is assumed natural and real. This puts the mind in an irrational place of exceptions, meaning that the exceptions will weigh more than the rules; sounds very ego centric. DNA is about 99.99% perfection in terms template connections, yet we fixate on 0.01% mutations to define reality. This may have some value for evolution and sickness, but not the preponderance of life.

This narrowing down to outside the preponderance of the data, is justified since math is assumed the foundation of reality. It is not seen as a language bridge used to help us see reality beyond our five senses. Scientists know the difference, but this same math foundation is also used by marketeers, politicians, bureaucrats, insurance companies, casinos, etc., that all benefit by faith in chaos being the rule. This saturate the social media worlds for reinforcement of the wrong mental bridge to life. It is bridge to death (used to model sickness, disease, extinction, etc.).

I hope to restore the sanity, since this deep neural firmware flaw, is behind so much of the world's problems; inertia of change and complexity based on exceptions instead of integration into simplicity. Complexity is why the US national debt is so high, and why the bogeymen rule fake news; scary exception to the rule become the foundation premise. It is an easy fix, but we need to see math as a language bridge, to peace or chaos, based on who drives; our foundation premises It can be the words of fake news and ambulance chaser lawyers, or common sense and the preponderance of all natural reality data. Your foundation premises will shape your world and our world; the driver.

Happy Easter.

 

[Polymath257]

As do ideal things.

I disagree. Ideals only exist in the mind, from all I can see.

They don't exist in reality, but there is a realm where necessarily existing things, that are not real, live - like forms and numbers. We don't see them with our eyes, but with our mind, that's why you might think they originate from there. But like real things, forms and numbers exist still, even if we don't "see" them at the time.

In what sense do they 'exist'? They are thoughts we have, which means they are in our minds.

Who is "we", Earthlings? At least the concept of numbers exists in non-human animals, too.

There is a huge difference between number *recognition* and having the concept of numbers.

For example, most people can look at a collection of 5 objects and immediately know there are 5 without actually counting. But very few can do the same with a collection of 26 objects. Other animals have shown number recognition, but I have yet to see evidence of them actually counting.

But there is another, more abstract concept of number. For example, suppose I have a bunch of sheep in my field. I 'count' them by having them go past and putting a stone in a jar for each sheep. I then use the stones in the jar as a representation of the sheep in the field. Instead of stones, i could use marks made on a tablet.

It is found *empirically* that this way of representing sheep (or other objects) is useful and consistent in many situations (excluding, of course, when sheep are born or die--in which case the representation can be extended).

An actual concept of 'number' goes one step more to say that the number of marks or stones is the 'same' as the number of sheep. This is a concept that we invented to describe certain situations where we want to keep track to make sure, say, sheep don't wander off.

Or it is evidence that Euclidean geometry is the "natural" way of shapes existing.

Except that there is no good reason to think that is the case. It is a good *approximation* in some situations, but I fail to see how spherical geometry is any less 'natural'. In fact, trigonometry was *first* applied to spherical triangles and only much later to plane triangles.

Basic math was initially done empirically. We can actually look at why the ancient Egyptians and Babylonians (a bad designation, by the way, but in common use) invented the concepts. We know from their writings what they were concerned about (keeping track of who put how much grain into the reserves, for example). We know that their calculations were done from empirical data because we have their techniques: and some of them are *wrong*, but good approximations, if we use Euclidean geometry. For example, the ancient Egyptians didn't have the 'correct' formula for the volume of a truncated pyramid.

Until the ancient Greeks invented the concept of 'proof', what we had was rules of thumb that gave good approximations for the things we wanted to keep track of. Often, even usually, those rules of thumb are seen as wrong using today's concepts. But they were good enough for the models they used.

 

[Polymath257]

Another aspect of this is that it seems to me that the discovered/invented dichotomy is a false one.

Would we say that language is invented or discovered? No person invented it, but it also certainly does not exist as some sort of ideal. The question itself seems to miss a great deal about how language works.

At the basic level, math is a language: it helps us describe aspects of the world around us and communicate them to others. It develops and changes over time as our goals and concerns change. It adds new concepts and new techniques as we explore and need new ways to express ourselves. Especially at the earliest times, it was mostly rules of thumb concerning how to be fair in division of goods or to keep track of how much each person contributed to the total. The rules were not selected as some sort of 'ideal knowledge', but rather as practical ways to ensure fairness.

It was only later that those rules were extended and became more abstract. Addition of 3 and 4 can be done empirically. For 3247 and 2937, some other techniques are required. So we invented those techniques. Over time, the notation simplified and the techniques became available to more people.

But I very much doubt that someone who was a hermit would independently discover the rules for decimal addition or multiplication. Too much goes into those rules.

It is also crucial to realize that many concept we see as 'trivial' or 'obvious' today were only invented/discovered quite late.

For example, the concept of a 'number line' is only about a thousand years old. The very notion would have been seen as self-contradictory to Greek mathematicians.

The concept of zero as a number is a bit older, but not by much. The ancient Greeks didn't even consider one to be a 'number'.

Algebra is another very late addition to mathematics. Prior to that invention, ALL mathematics about numbers was completely language based: problems were described using ordinary language and the solutions were similarly described. general techniques were usually not given. Instead, a collection of similar examples were worked out to show the pattern.

Historically, math developed out of language as a way to describe things about the world around us. it changes and develops over time in ways that, I think, negate the claims of its universality. There have been too many *arbitrary* choices and distinctions made throughout time.