Why does math work in our reality?

[apeiron]

Ok
-what about the Ptolemaic system? - a great pattern fit. It was considered to be dogmatic, axiomatic, rigid and false because it did not penetrate our ideas of how God the geometer would have constructed the universe.

Epicycles did kind of fit with a harmony of the spheres. But we now think it ugly because it depended too much on construction - the addition of cycles - and not enough on global constraints (such as satisfying a universal law of gravitation).

At an instinctive level, we have long known that "good modelling" is about a natural balance of construction and constraint, local atoms and global laws. Now is the time to make this formally explicit in the form of an equilbration principle. Which is the main thing I've been working on with my interest in vagueness, dichotomies and hierarchies.

- Einstein's theory of relativity originally did not,at first, explain new data - it discovered a new concept of space time - physicists believe that is was a totally unexpected and anomalous idea precisely because of this. They agree that physics could have gone along explaining known data just fine without it.

Michelson–Morley? Mach and centrifugal force?

There is a deep book on this subject written by Henri Poincare called Science and Hypothesis. In it he says that there are two schools of physics, the English school which denies the need for hypotheses about the nature of reality and believes that all science is just pattern fitting and the Continental school which says that underlying assumptions/hypotheses are necessary. the pattern fitters were notably Newton and Faraday both who explicitly claimed that they needed no hypotheses - maybe Maxwell also though I am not sure of his view on this. Curiously it seems that there were really two schools of physics centered around this controversy.

One school worries about the information that must be discarded in modelling, the other doesn't.

Well actually the Newtons and the rest usually do wonder about the gap between reality in its fullness and their reduced descriptions that involve things like action at a distance.

But modern epistemology - Rosen's modelling relations being the best articulation I have come across - does away with this old hangover.

 

[apeiron]

I disagree that math is a science, so I see no problem in asserting that science is modeling while math is not. Math is a tool used by science to create models. Mathematical theories, in a way, can be said to be models themselves, they just aren't models of anything in particular without science attached. Math is the logical analytic extension of axioms or assumptions.

The thread of prejudice running through your argument here is that knowledge is passive - it "exists". Whereas I am arguing from the opposite position that knowledge is active - it is about doing things, indeed getting things done. So that is why "modelling" is the chosen word. We do no represent reality or behold reality, instead we are seeking to have control over it - even if it is simply control over our perceptions at times.

Yes, you can talk about maths as people creating axioms and then investigating all the patterns that can flow from the axioms. This describes the day-to-day for many academics. It seems a very passive and interior exercise. And often is sterile. But the maths that gets sociologically rewarded is then the maths that turns out to be useful for control over the world, so betraying its true purpose.

So as I say - based on modelling relations epistemology - there is a natural divide into models and measurements. An observer needs the general of his ideas, the particulars of his impressions. And psychology tells us how these two develop from vague to crisp through their mutual interaction. The way a newborn baby learns to make sense of its world through active exploration.

This natural division is then repeated in our formalised disciplines. We have a method for constructing models, a method for making measurements. Maths is about fashioning tools for model construction. It may involve philosophy too in developing its crisp axioms.

 

[apeiron]

Kantian innateness is essentially a procedure-based innateness, consisting in an a priori active readiness of the mind for implementing rules of synthesis, as opposed to the content-based innateness of Cartesian and Leibnizian innate ideas, according to which an infinitely large supply of complete (e.g., mathematical) beliefs, propositions, or concepts themselves are either occurrently or dispositionally intrinsic to the mind. But as Locke pointed out, this implausibly overloads the human mind's limited storage capacities.​

In other words, logic is innate, ideas are not. I like Bertrand Russell's treatment of the issue. Russell's laws of thought are somewhat similar to Kant's categories of perception. He talks about universals and relations in http://www.ditext.com/russell/russell.html chapters 7-10.

.

Yes, neurology tells us that the brain indeed has a "logic" - a way of arriving at a crisp local orientation to a global world. And that process is dichotomisation. Figure-ground, focus-fringe, attention-habit, conscious-preconscious, etc.

And this "real logic" has scale. There is always a local-global asymmetry involved. Global universals and their local particulars. Whereas modern symbolic logic has developed through the discard of scale - the reduction of asymmetry to (mere) symmetry. So the yes/no, on/off, binary and scaleless choices of information theory.

 

[wofsy]

I appreciate the fact you like a good argument but you have to at least make an attempt to understand what the other side is saying. Saying you find things meaningless, or you don't believe, is not a very interesting position to hold. So I'll limit my responses.

"real mathematics is a way of discovering our ideas of space, geometry and number"

You mean pattern and form. Self-consistent organisation.

"What empirical model of reality would you say the theory of represents?"

As you say, it all starts with observation of the world - modelling at its most natural level. The ancients noted the regularities of the world and generalised to create some structural truths about objects like triangles and morphisms like spirals.

Then mathematics has developed by even greater generalisation - as articulated by category theory for example. And demonstrated in the move from euclidean to non euclidean geometry, or simple to complex number. We find that we exist in a world that is highly constrained (3D and flat - and scaled) and then generalise by successively removing those constraints to discover if self-consistent regularity still exists.

And you can see what happens when you do this with quarternions and octonions. The regularity frays. Properties like division erode as you generalise the dimensionality. Instead of producing crisp algebraic answers, the meaning of the algebra becomes vague.

So the most general maths can cease to model crisp properties that were there in the original "empirical" view.

When this happens, many say maths has just wandered off into the wilderness - or a landscape in string theory's case.

But my philosophical approach is different. I am saying that generalisations lead back to vague potential. And the way to rescue the situation is by also building the global constraints - the selection rules that represent the idea of "self-consistency" - back into the maths explicitly. So maths with scale.

"Science develops because people question or ideas of reality not because we model it."

Ideas are models - ideas formalised.

I apologize for saying things were meaningless - but your use of language I found inpenetrable -plus you took a lecturing tone. So I felt I was being lectured to with non-specific vague words. This was a sincere reaction and I felt very frustrated.

I still don't exactly know what your language means and that is why i gave up. I think clarity of expression is necessary and lack of clarity is a sign that the person does not know what they are talking about.

For instance you just told me what I mean - as if I don't know what I mean. That is condescending. I have no desire to fight with anyone and I am totally open minded. But you have not been clear as far as I am concerned and you have been lecturing. That is also not very interesting.

 

[apeiron]

I think clarity of expression is necessary and lack of clarity is a sign that the person does not know what they are talking about.

.

I accept I can be irritating. All I can say is that I was also feeling irritated.

Also I believe that any lack of clarity is due to the unfamiliarity of the ideas I am attempting to communicate rather than my alleged deficiencies as a communicator.

Yes, these ideas I am expressing do indeed come from a different community - a rather small band of systems thinkers such as Salthe and Pattee. And I understand how opaque they can seem. It took many years of discussion for me to come round to some of them. And we are also talking about work in progress - current research.

Anway, I have tried to create introductions to some of the key ideas like Vagueness - see this thread.

https://www.physicsforums.com/showthread.php?t=301514&highlight=vagueness

 

[wofsy]

Ok So now that we realize that we are all sincere and serious here and are not trying to be dogmatic and contentious it would greatly interest me to understand how you view some of the examples that I suggested. Why not start with the examples of fields of mathematics that do not arise from attempts to explain empirical data. A simple one that would could all talk about without taking a math course first might be the discovery of hyperbolic plane geometry in the 18'th century.

My understanding is that people for centuries felt that Euclidean geometry was intrinsic to the idea of geometry itself - that the parallel postulate was indispensable to the idea of space.

I believe that Kant event thought that Euclidean geometry was an a priori synthetic idea meaning that it was not an empirical model but rather an intrinsic feature of our experience of spatial relations ships. But like logic which you seem to agree is intrinsic, Kant thought that Euclidean geometry was intrinsic.

Many others agreed with him and realized that if this were really true then the parallel postulate should be provable from the simple axioms of space that describe the way lines intersect and how they separate a plane. One axiom said that two points determine a line. Another said that a line separates a plane into two half planes. A third said that two lines in a plane can intersect in at most one point.

These guys already knew that parallels must exist - not empirically because that would be impossible to test - because they knew that two lines that intersect a third at right angles must be parallel. They just couldn't prove that they were unique. It was uniqueness that got them.

This lead them to question their intuition/picture of straight really meant. gauss finally came up with a model of plane geometry where lines were actually curves and where the parallel postulate failed. In his geometry there were infinitely many parallels through any point.

After that people thought that there were two possible intrinsic geometries of space and only after they realized this did they actually try to test it out - under the assumption of course that our picture of space that is derived from sense experience actually must obey geometrical laws. Gauss went out and measured large triangles on the Earth to see he he could detect angle defects away from 180 degrees.

So you need to take this Kantian or perhaps Platonic - you would know better than I - way of looking at things and tell me how it was only just discovering empirical relationships - generalizing observations - through models. This to me, and I know for sure for gauss and his colleagues - was an investigation into the intrinsic nature of our ideas of space. The empirical modelling part was not central to the investigations and came afterwards when Gauss realized that if one believed - by either philosophy or faith - that experience actually exhibits the laws of geometry that one should then be able to test for the two possibilities.

Let's make this the starting point and take this paragraph as a first step to get thing going.

 

[apeiron]

My take here starts by saying it is a false dichotomy to think the situation would be EITHER empiricism OR platonism (or constructivism or intuitonalism, or however else we want to phrase this traditional divide between "looking out" and "looking in"). Instead - logically - it must always be BOTH. As the complementary extremes of "what can self-consistently be".

This is what happens because I chose asymmetric dichotomisation as the foundation of my logic. This is of course the unfamilar bit, even though it starts from ancient greek metaphysics (Anaximander, Aristotle), was messed about a bit by the likes of Hegel, and reappears in modern times with Peirce.

Now asymmetric dichotomisation says that any (vague) state of possibility or potential can only be (crisply) divided if that act of separation goes in two exactly "opposite" directions. And by opposite, this is not symmetric as in left/right or other kinds of symmetry breakings which have just a single scale. It must be an asymmetric breaking that is across scale and so results in completely unlike outcomes (as opposed to merely mirror reflections of the same thing).

If you are with me so far, then the classic examples of asymmetric dichotomies in metaphysics are local-global, substance-form, discrete-continuous, stasis-flux, chance-necessity, matter-mind, vague-crisp, subjective-objective, atom-void, space-time, location-momentum (and the list goes on, but these are among the "strong ones").

You can see that each is both the very opposite of the other, and yet also logically mutual or complementary. That is because each is defined actively as the exclusion of the other. Pure substance would be a stuff that has absolutely no form, and form is that which has absolutely no substance. (Even Plato had to have the BOTH of the forms and the chora).

So this is an emergentist and interactions-based logic or causality (a logic being a generalised model of causality in my book). You cannot have one side arise into being, into existence (or persistence) without also forming the other. As one arises (in thought or reality) by becoming everything that the other is not.

As I say, Anaximander was the first to articulate a vagueness => dichotomy => hierarchy approach to modelling causality, the logic of reality. Aristotle then polished it up (as in the law of the exclude middle). Today, you can see mathematical sketches of the idea in the symmetry breaking models of condensed matter physics, in hierarchy theory, and even in some basic stabs at maths notation.

Check out Louis Kauffman's musings on this...
http://www.math.uic.edu/~kauffman/Peirce.pdf

The laws of form are another stab...
http://en.wikipedia.org/wiki/Laws_of_Form

A gateway to Peirce's writings (which are only a precursor to what I'm talking about)...
http://www.cspeirce.com/

And others currently treading some of the same ground (though I would have many criticisms of Kelso's actual approach)...
http://www.thecomplementarynature.com/

Anyway, I hope you can appreciate that this is like swapping in, swapping out, a complete computational architecture. There is standard logic based on atomism, mechanicalism, locality, and other good stuff which is like your classic sturdy von Neumann serial processing engine. It works, no question. Then over here in left field, there is an attempt to build an architecture of thought, a way of modelling, that is founded on very different basic computational principles. It is like the attempt to get neural networks off the ground. Some kind of global, holistic, hierarchical version of logic. And while it looks promising, it is still a long way from commercialisation.

But anyway, let's take these still developing ideas and apply them to the question you asked.

Again, for me on the grounds of logic (all reality always works this way) I would come with the expectation that the story is going to be not either/or but instead both, and interactionist. So yes, strong dichotomies always emerge, and then the whole point is that they emerge because their existence is self-consistent in the wider view. They are mutually causal, or synergistic as asymmetric extremes.

Therefore it does seem that the creation of mathematics has this basic divide. There is either the pure development of ideas, or the discovery of ideas from observation. And my logic would force me to expect a mutually emergent story. The firming up of ideas inside a person's head allows them to make more detailed observations of the world, which in turn allow for more development of ideas inside their head. And these two parts of the action are driving each other ever further apart in scale. As the observations get ever smaller, ever finer, ever more particular, so the ideas get ever more general, ever more global and universal, ever more lacking in picky detail.

Now to take the specific example of non-euclidean geometry. The tale of the discovery follows this dichotomous logic. At first, forms got separated from substances in a way that divided the flat 3D world of immediate experience. Then as mathematicians realized that just three dimensions is a rather particular choice, and likewise just flat space was a rather particular choice, they could make a leap of generalisation to allow infinite dimensionality and any curvature. Their ideas became less particular, and so more general.

At the same time, this step in one direction brought with it a matching step in the ability to make ever finer "observations". It became possible to model some world with some particular curvature or number of dimensions. Maths could start exploring imaginary worlds of any crisply chosen design (and science could then use this new technology to test our actual world against the new variety of predicted designs).

So dichotomisation is the logic by which humans stepped back to see more. And then I would go further - from epistemology to ontology. Dichotomisation also is how the world probably actually emerges.

Taking non-euclidean geometry, we can see for example that "flat space" is precisely the average, the sum over histories, of curved space. If you have a dichotomous spectrum from purely locally hyberbolic space (disconnecting sea of points) to purely global hyperspheric space (curvature which makes a continuous or perfectly closed space) then flatness is the average, the equilibrium outcome, of these extremes "in interaction".

Of course this is still a hypothesis as I'm not sure how to go about constructing a mathematical proof of the idea. But I am just sketching the kind of answer I would expect to be the case if dichotomous logic is a valid logic.

There is another argument about why there would be just three spatial dimensions. But I can save that for some other time as it is even more left-field if Peircean semiotics is unfamiliar terrain.

To sum up, all my arguments stem from applying a different computational architecture. And it is not an arbitrary choice as - dichotomously - there would have to be exactly two deep models of logic/causality. Standard logic is one pole, and now I am working with people in developing the other pole. I see this as great news for good old fashioned atomistic logic as it cements its authority in place. It can be "right" because there is also the asymmetric view now making it "right" - that is, together they exclude the middle, all other possible approaches to logic.

So dichotomies rule. And the division over whether maths is derived from intuition or perception is a classic example of how both in interaction, creating a virtuous spiral of development, is the answer.

Then the logic of our minds is also the logic of reality itself. Dichotomies or symmetry breakings are also how things happen "out there" - how systems develop into being, complex hierarchies arising out of vaguer potentials because they are the self-consistent way a vagueness can be stabily, self-persistently, divided.

I am sure this is still indigestible. But just focus on some dichotomy and see for yourself if you can break it down differently.

Local-global is the most fundamental dichotomy I believe - pure scale. Though (dichotomously) it is then paired with an equally fundamental dichotomy vague-crisp. One talks about what exists, the other how what exists has developed.

But substance-form is the Athenian set-piece debate. Or you could back up a bit to consider the weaker dichotomies of stasis-flux or chance-necessity or atom-void.

 

[wofsy]

I am digesting your words - thinking about them - will reply when I have something cogent to say.

 

[apeiron]

these might also be useful...

A view on Rosen's modelling relation
http://www.osti.gov/bridge/servlets/purl/10460-5uGkyu/webviewable/10460.pdf

A review of his book
http://www.metanexus.net/magazine/ArticleDetail/tabid/68/id/2589/Default.aspx

systems biology - the project
http://www3.interscience.wiley.com/cgi-bin/fulltext/116833284/PDFSTART

 

[emyt]

The entire argument is based on correspondence to observation. There is nothing special about math in this sense. It needs to be corroborated. 2 humans plus 2 humans can equal 5 if one gets pregnant. Therefore 2+2=5. It is just generally true, when you categorize things together in groups that 2+2=4.

Math is about generalizations, that is where its strength lies. But there is no math statement that stands on its own. Its correspondence to observation is what makes it a valid generalization. Observation is what math is built on.

If 2+2 = 5 because one gets pregnant and thus creating another human being, the equation would be 2+3 = 5; unless you don't consider the baby inside the womb to be a human being, but then why would you at the same time say that 2 humans plus another 2 humans equal 5?


-----------------------


Basic arithmetic comes from our concept of units and quantity. 1 + 1 = 2 is simply 1 full unit of what we are considering, plus another full unit, equals 2 of those units. Other more abstract concepts can be deduced from previously deduced logical principles, we can't expect the Engilsh language to be capable of encompassing any idea there could ever be. Not all propositions refer to the most fundamental logical principles (but they could be deduced from other fundamental logical principles)

 

[JoeDawg]

If 2+2 = 5 because one gets pregnant and thus creating another human being, the equation would be 2+3 = 5; unless you don't consider the baby inside the womb to be a human being, but then why would you at the same time say that 2 humans plus another 2 humans equal 5?

Why would you? well that is the point. The kind of system you create will depend on what you want to do with it. If you want to describe procreation as part of your system, then 1+1=3, and 2+2=5, etc...

Your math depends on correspondence. There are mathematical systems that don't include the concept of zero. There are systems that are based on one, a few, more than a few.

And this is why you literally have to create new math to describe experiene which is quite alien to what you normally experience, for example, quantum reality. Because different things are important to you.

Saying, why, and what is important, is just part of defining your system.

 

[Sorry!]

Why would you? well that is the point. The kind of system you create will depend on what you want to do with it. If you want to describe procreation as part of your system, then 1+1=3, and 2+2=5, etc...

Your math depends on correspondence. There are mathematical systems that don't include the concept of zero. There are systems that are based on one, a few, more than a few.

And this is why you literally have to create new math to describe experiene which is quite alien to what you normally experience, for example, quantum reality. Because different things are important to you.

Saying, why, and what is important, is just part of defining your system.

Actually I'm pretty sure it would look like this :

M⋃F → B(male union with female leads to baby or 1 union with 1 leads to 3)?

 

[JoeDawg]

Actually I'm pretty sure it would look like this :

M⋃F → B(male union with female leads to baby or 1 union with 1 leads to 3)?

Except when it doesn't.

You could also describe it:

1+1= 2 1/2

Or you could add up the cells in each body.

Or mathematically represent the genetic information.

Its all about what you consider relevant. Abstract reasoning allows you to make whatever distinctions you like. This is why math can be very precise. The only limit is your assumptions, and what you consider relevant, that is, what the math corresponds to.

 

[disregardthat]

If, in your system, get 2+2=5 then what you are adding are not numbers! As a consequence of the properties of numbers, 2+2 always equal 4, and hence if 2+2=5 you are not dealing with numbers. This is not a flaw in the mathematical system, but in your ability to properly model the situation with mathematics. If this is a consequence in your system, then, clearly, the total amount of human beings cannot properly be computed by adding.

 

[JoeDawg]

If, in your system, get 2+2=5 then what you are adding are not numbers!

They are numbers, they just don't correspond to the same 'things'. Which is the point.

This is not a flaw in the mathematical system, but in your ability to properly model the situation with mathematics.

Its not a flaw, its a different model.

If this is a consequence in your system, then, clearly, the total amount of human beings cannot properly be computed by adding.

And how do you confirm this?

 

[disregardthat]

They are numbers, they just don't correspond to the same 'things'. Which is the point.

No. As a consequence of the definitional properties of numbers 2+2=4 is always correct. 2+2=5 is false statement when dealing with numbers, and thus is the statement is correct, they is not numbers.

This is not a flaw in the mathematical system, but in your ability to properly model the situation with mathematics.

Its not a flaw, its a different model.

Well, a model with dissatisfactory accuracy, which was the point...

If this is a consequence in your system, then, clearly, the total amount of human beings cannot properly be computed by adding.

And how do you confirm this?

Obviously by observing that 2+2=5 is a result in the system, which is untrue for numbers.

 

[JoeDawg]

Obviously by observing that 2+2=5 is a result in the system, which is untrue for numbers.

Sigh.

And where does the definition of 'numbers' come from?

 

[disregardthat]

Sigh.

And where does the definition of 'numbers' come from?

Try peanos axioms for example. And ZFC defines numbers based on the concept of sets. This is a rigorous construction of the natural numbers.

 

[disregardthat]

Math works in our reality because we define our physical concepts in mathematical terms. Mathematics is an extremely effective tool for describing physical theories for exactly this reason. Because physical laws seem to follow certain laws, we are naturally encouraged to apply our mathematical concepts too it, and with great accuracy.

 

[JoeDawg]

Try peanos axioms for example.

And what is an axiom?

Its an assumption.

Why would we choose one axiom over another?

Because some axioms have a broader scope, they describe a wider range of experience.

Math is about generalizations, we draw these generalizations from observing consistency in the world, the greater the scope of the axiom, the more we can use it, and the more we do use it.

1+1=3 is useful, if you are talking about sex. But its not useful when you are discussing apples. 1+1=2 is useful for apples, as well as a large range of things we as humans consider relevant.

 

[disregardthat]

And what is an axiom?

Its an assumption.

Why would we choose one axiom over another?

Because some axioms have a broader scope, they describe a wider range of experience.

Math is about generalizations, we draw these generalizations from observing consistency in the world, the greater the scope of the axiom, the more we can use it, and the more we do use it.

1+1=3 is useful, if you are talking about sex. But its not useful when you are discussing apples. 1+1=2 is useful for apples, as well as a large range of things we as humans consider relevant.

An axiom is not an assumption as in the context of "taking it for granted". The concept of an axiom is a definition. When we are postulating an axiom, then we are defining whatever we are talking about. We are not talking about something we think we know something about, and then saying something we might think is true, and then take it for granted. An axiom is an assumption made in order to explore the consequences, and this is a critical point.

Math is about making generalizations, but it does not base itself upon empirical evidence, although it is inspired by it. Mathematics is (luckily) based upon rigorous definitions, which make silly statements like 1+1=3 meaningless if you are talking about numbers.

 

[JoeDawg]

although it is inspired by it.

Inspired? What does that even mean?

which make silly statements like 1+1=3 meaningless if you are talking about numbers.

I think I see the problem here, you've decided that certain axioms of math have some sort of Platonic existence. But what 'numbers' are, is whatever they are defined to be. Now, some definitions are more useful... empirically, and those are the ones we keep, use, modify, and refine. However, it is via observation that we decide which axioms are useful, and which are meaningless. You can't generalize from nothing, first you have to have instances, and then you develop rules based on those instances... this is how logic and math work. Oh, and your patronizing tone is actually quite amusing. I don't disagree with most of what you said, I just don't think it means what you think it does.

 

[disregardthat]

Hehe, I think we have lost track of objectivity here. This debate has obviously come to a halt, I guess we have to agree to disagree. I won`t discuss anything but arguments.

However, I will say this: It doesn`t matter how we are choosing our axioms here, what is important is that we follow those we have chosen. In any reasonable definition of numbers, (read: axioms), 2+2=4.

I am patronizing?

Sigh.

And where does the definition of 'numbers' come from?

 

[JoeDawg]

In any reasonable definition of numbers, (read: axioms), 2+2=4.

And reasonable means... whatever Jarle agrees with.

Like I said... patronizing.

 

[disregardthat]

And reasonable means... whatever Jarle agrees with.

Like I said... patronizing.

You aren`t really discussing, are you?

 

[apeiron]

And reasonable means... whatever Jarle agrees with.

Actually Jarle has been taking the more reasonable line here. Yes the idea of numbers may be a generalisation from experience, but it is also a maximally general one. As far as we can know. From the prime test, which is the self-consistency of the algebraic structures we find we can spin from the number system.

The "1+1=3 when we are talking about making babies" is a childish debating point. It is a description using numbers to talk about some specific kind of biological event. It is not a self-consistent consequence of the numbers themselves.

I certainly believe that we model reality. And also that our concept of number can be challenged. Axioms are always questionable.

But it becomes just silly to not understand that axioms are generalisations that can then have matchingly crisp or definite consequences. So you can't just try to assign your own private meanings to the objects of that system of morphisms as JoeDawg wants. The actual model consists of both its axioms and its consequences.

As ever, you have to keep your eye on the dichotomies at the centre of these things. That is why maths moved on to category theory in its search for its fundamental ground. Structure-preserving change, patterns or symmetries that can persist.

 

[JoeDawg]

The "1+1=3 when we are talking about making babies" is a childish debating point. It is a description using numbers to talk about some specific kind of biological event. It is not a self-consistent consequence of the numbers themselves.

I was never advocating that is was a replacement for 1+1=2, nor that we should throw out mathematics.

The point was simply that we derive our math from experience. The reason it is self-consistent is because we have specifically created it to be so, and moved from using numbers that only really work in specific cases, like with babies, to number systems that cover a wider amount of instances and with great precision. 1+1=3 may be descriptive, but only in a limited sense. 1+1=2 is also limited, however, integers are not as precise as decimals. And fractions also function differently. We have found ways to convert from one to the other, but each has problems and limitations.

And as we add more to math we specifically seek ways to make it consistent with what we already have. Its only self-consistent because that's the standard by which it is judged useful, and usefulness is about correspondence to reality.

 

[slider142]

If you dislike the babies argument, consider a universe where Bose-Einstein condensates were more common than in ours. In that universe, the development of mathematics may have favored axioms that result in the theorem 1+1=1 versus the more exotic 1+1=2 branch (if anthropomorphic consciousness is even possible in such a universe).

 

[wofsy]

I was never advocating that is was a replacement for 1+1=2, nor that we should throw out mathematics.

The point was simply that we derive our math from experience. The reason it is self-consistent is because we have specifically created it to be so, and moved from using numbers that only really work in specific cases, like with babies, to number systems that cover a wider amount of instances and with great precision. 1+1=3 may be descriptive, but only in a limited sense. 1+1=2 is also limited, however, integers are not as precise as decimals. And fractions also function differently. We have found ways to convert from one to the other, but each has problems and limitations.

And as we add more to math we specifically seek ways to make it consistent with what we already have. Its only self-consistent because that's the standard by which it is judged useful, and usefulness is about correspondence to reality.

Though I have been trying to stay out of this until I understand the philosophical points you are making my gut still objects to the idea that reality and mathematics are somehow disjoint. It seems to me that pure sensation has no intrinsic structure. So to say that you are generalizing from something without structure seems impossible. Controlled observation can only give us clues to the structure of reality. But reality is not something we directly experience or observe. Though a fusion of thought and observation we lift the veil of sense experience.

 

[apeiron]

If you dislike the babies argument, consider a universe where Bose-Einstein condensates were more common than in ours.

If your reality were a Bose-Einstein condensate, where would the notion of a this one, as distinct from that one, derive? It would seem you would only feel like saying 1=1 at best? Experience would not yield a 1 plus a 1.