Why does math work in our reality?

[apeiron]

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.

So you are arguing against yourself here. It seems there was some innate and inevitable trend to be discovered. A path that leads from the vaguely useful to the crisply useful, from the particular to the universal.

Of course, human civilisation did not actually start with a mathematics based on babies and then progress to something better.

Psychologically, the first and most natural dichotomy was probably the distinction between the one and the many. Or figure and ground, event and context, signal and noise. The idea of symmetry and then the symmetry breaking.

And anthropologically, if we want to focus on utility, the origins of maths probably had most to do with the cycles of the days and the seasons. Cycles of death and renewal. So more geometry than algebra. Though perhaps they did notch off sticks to count off cycles of the moon.

Counting became important in ancient agricultural civilisations with hierarchical ownership. Counting boards and tally sticks to keep track of the goats and sheafs of wheat. But I don't think even the Summerians recorded 1 goat + 1 goat as making 3. Or derive from that the further truth that if I have 3 goats and give you 1, then that must leave me also with only 1.

Again, mathematical systems must follow a certain path - the dichotomy defined by category theory. You must have the fully broken symmetry of the local and the global, the one and the many, the object and the morphism. Yes this is derived from experience - and also appears to be a truth about reality. Which is why maths works.

The mistake you keep making is then to just focus on one half of the dichotomy, of the broken symmetry. The number 1 does not stand alone. It is defined only in relation to its context. Which is why 1 has a stabilised meaning and cannot float free as something that could be defined anyway we choose.

Of course there is then a further epistemological wrinkle to all this. Out there in reality, symmetries are not truly "broken". Instead the breaking apart is merely approached in the limit. However in maths, as a modelling choice, we do treat symmetries as properly broken. So we treat the number 1 as not the limit of the act of separating the one from the many, but as actually - axiomatically - a thing which is separate, isolate, discrete. So maths is in fact unreal in this crucial regard. It appears to say something about reality which cannot in fact be.

 

[JoeDawg]

It seems to me that pure sensation has no intrinsic structure.

In as far as its a function of biology, I would say it does, but I'm not sure what you mean here. The human mind instintively separates experiences into events and objects.

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.

Well, I would say reality is what we experience, both with regards to thinking and observation. The source of reality is the mystery.

 

[JoeDawg]

It seems there was some innate and inevitable trend to be discovered. A path that leads from the vaguely useful to the crisply useful, from the particular to the universal.

Utility is relative. Consistency in experience gives us the foundation. The direction and scope are up to us. Narrowing the scope when needed just gives you a better picture of what you want to see... like a fractal.

Of course, human civilisation did not actually start with a mathematics based on babies and then progress to something better.

It would have been more basic than that, but the arbitrariness of the example is also important to my point. I think many people want there to be some ultimate math, like a ToE, but math is what it is used for.

And anthropologically, if we want to focus on utility, the origins of maths probably had most to do with the cycles of the days and the seasons. Cycles of death and renewal. So more geometry than algebra. Though perhaps they did notch off sticks to count off cycles of the moon.

The ancient Egyptians invented geometry to deal with the problem created by the Nile flooding the land. It was good for the land, but it made it difficult to allot farmland. The flooding destroyed all landmarks. Geometry solved this problem. It was also useful with regards to astronomy, and the building of tombs. But no perfect circles exist.

Again, mathematical systems must follow a certain path - the dichotomy defined by category theory.

One doesn't need math to have categories or dichotomies. Math just formalizes what our minds and bodies already do.

The mistake you keep making is then to just focus on one half of the dichotomy, of the broken symmetry. The number 1 does not stand alone. It is defined only in relation to its context. Which is why 1 has a stabilised meaning and cannot float free as something that could be defined anyway we choose.

I agree there has to be context, which is why I used procreation as context. It's simplistic and not broadly useful, but it makes the point. Without context all math is just squiggles on a page.

Of course there is then a further epistemological wrinkle to all this. Out there in reality, symmetries are not truly "broken". Instead the breaking apart is merely approached in the limit. However in maths, as a modelling choice, we do treat symmetries as properly broken. So we treat the number 1 as not the limit of the act of separating the one from the many, but as actually - axiomatically - a thing which is separate, isolate, discrete. So maths is in fact unreal in this crucial regard. It appears to say something about reality which cannot in fact be.

The map is not the territory.

 

[wofsy]

In as far as its a function of biology, I would say it does, but I'm not sure what you mean here. The human mind instintively separates experiences into events and objects.

Well, I would say reality is what we experience, both with regards to thinking and observation. The source of reality is the mystery.

biology is an explanation of experience - there may be other explanations - these are not what experience is - if you are saying that all ideas generalize experience then from this point of view you are just making a statement about biological theory.

Different people attach different intellectual constructs to experience of the outside world. Does this mean that they have different instincts? Does this mean that there are multiple realities?

The Impressionist era artists, particularly, Monet and Cezanne, tried to eliminate intellectual constructs from their images. They rejected the idea that such things as perspective and the theory of light actually are part of experience. They viewed these things as intellectual overlays. Their goal was to record experience at the moment of sensation just before the division into objects occurs. This is what Cezanne meant when he said that when he paints he tries to learn from nature. This is what I meant by unstructured experience. This is why their pictures often appear flat and are not supported by a geometrical skeleton as in classical art. At that time, people believed that pure experience was individual and even racial. from your point of view, they would say that there are as many realities as there are individuals. And I get the impression that this is what you are saying.

To me, whether experience is instinctively organized or not - and I am not sure how anyone knows this - that does not mean that experience is not separable into is cognitive constructs and sensory elements. To me the intellectual constructs are those things which are not experienced and the sensory data is what is given - that which is directly experienced.

 

[S.Vasojevic]

Well I think it is because our world is about relations between things. Why object that has mass of 1 kg has twice smaller mass then object of 2 kg? If you say it has four times smaller mass, what would you say comparing it to object that has mass of 4 kg? Nature of course does not care about numbers, they are symbols made by us in our modest attempt to explain our existence through relations between entities.

 

[JoeDawg]

biology is an explanation of experience - there may be other explanations - these are not what experience is - if you are saying that all ideas generalize experience then from this point of view you are just making a statement about biological theory.

It seems to be the case that Ideas are distinct from Experiences.
And it also seems to be the case that certain Ideas are linked to certain Experiences.
Experiences however, seem to have more detail and specificity. Ideas seem more vague.

If you want to accept the radical empiricism of Berkeley and state that matter doesn't exist and such... yes, that is another 'explanation'... but accepting the conceit of the common understanding of modern science and physicalism, yes, all ideas generalize experience.

Different people attach different intellectual constructs to experience of the outside world. Does this mean that they have different instincts? Does this mean that there are multiple realities?

Defining reality can be confusing. Often people equate reality with existence, which is fine, but it ignores the fact that experience sometimes contradicts existence. At which point you have two 'realities'. Dreams are a good example of this. Last night I flew through an alien city... does the alien city exist the way my computer does... well no, but it does, or did, exist.

So yes, in that sense everyone has their own 'reality' following them around. And those realities all seem to come from the same source... existence.

As to instincts... there are biological instincts and learned instincts, so yes different people can have different instincts. An example of the latter instinct would be something you are trained to do, and therefore do automatically without thinking.

At that time, people believed that pure experience was individual and even racial. from your point of view, they would say that there are as many realities as there are individuals. And I get the impression that this is what you are saying.

In terms of experience sure, but not ontologically. Its true we can't know for certain whether 'its all just a dream', but this is where being reasonable comes in. I see no reason to doubt that there is a common source for experience.

To me, whether experience is instinctively organized or not - and I am not sure how anyone knows this - that does not mean that experience is not separable into is cognitive constructs and sensory elements. To me the intellectual constructs are those things which are not experienced and the sensory data is what is given - that which is directly experienced.

Problem there is 'true sensory experience' is not self-reflective. Think of how animals and babies seem to live in the moment. Any analysis of experience, and certainly taking the time to paint it on canvas, would entail cognitive constructs. Even talking or thinking about an experience... puts it within a constructed framework. Meaning is cognitive. Experience just is, and then it's gone.

 

[apeiron]

To me the intellectual constructs are those things which are not experienced and the sensory data is what is given - that which is directly experienced.

The idea that experience is given - the ineffability of qualia - is not something that would be supported by psychology and neuroscience. Experience is also constructed.

The terms I prefer to use here are "ideas" and "impressions" as it helps preserve the constuctedness of both aspects of awareness. One is not being favoured over the other in term of veridity (or lack of it).

Now what is the nature of the actual divide (I mean dichotomy) that you are sensing here? The instinctive separation you want to make?

It is between the general and the particular, the model and the measurements. Or as in Grossberg's neural nets, the long term and the short term memories.

So in all these ways of saying the same thing, we have something that acts as the longer lived context - the idea that constrains. And then we also have the moment to moment impressions, the fleeting train of events, that constructs some particular state of experience.

And the two scales of mental activity are in interaction. They are not separate processes but instead separate levels of process.

So ideas are the established habits of memory, anticipation and thought which serve to give shape to impressions. They make sense and organise each moment. Awareness is created top-down.

But equally, impressions over time build up the ideas. The brain learns by generalising from what it thinks happened (the experiences it constructed) and so builds broader, sturdier, habits of interpretation/perception.

So all this activity is subjective. There is no direct objective access to reality. But, being a systematic approach, the subjective collection of ideas~impressions does come to model reality very well for our purposes.

And then maths/science/philosophy are activities that try to repeat this basic cognitive formula on a still broader social scale. Societies have purposes and evolved models of reality that serves them.

 

[JoeDawg]

The idea that experience is given - the ineffability of qualia - is not something that would be supported by psychology and neuroscience. Experience is also constructed.

While this is arguably true, its also more complex than that...
Experiences and Ideas exist on the level of consciousness. They are, in a sense, immediate.
Neuroscience and psychology work on the level of explanation.
So you can indeed describe experience as unconstructed. Its only after experience has been assessed and compared that we get the sciences, and then we work backwards for an explanation. It is then that we can view experience as constructed.

Which is not to say that neuroscience and psychology don't offer good explanations, but they are heavily dependent on that 'ineffable qualia', which is our primary mode of being.

Ideas are more obviously constructed, since they involve internal (mental) processes, not external sources.

 

[apeiron]

So you can indeed describe experience as unconstructed. Its only after experience has been assessed and compared that we get the sciences, and then we work backwards for an explanation. It is then that we can view experience as constructed.

I understand what you mean but this is what I call the introspectionist fallacy.

In fact paying attention to your experiences is a highly artificial and learned skill. Animals and babies can't do it. And it takes a lot of practice and scaffolding for even modern Western adults.

You can argue that there are degrees of construction. So the naked sensation of redness is perhaps less obviously mediated than your perception that a ship on the horizon is a large object a long way away.

Yet still the very act of stopping and contemplating "redness" is a highly constructed - and constructing - action. Your brain has to suppress attention to much else to manage to make it seem like the redness of something red is filling your awareness.

Or to use a better example, think of the tricks that an impressionist painter goes through to see the distant hills as purple not brown or green. Cut out a little square in white card and hold it up to physically block out the contextual information that is fooling your appreciation of the pure actual colour.

Yes, some things may be less mediated, less apparently constructed. But in the end, all experience is the result of some act of mediation, some constructive effort and not about naked witnessing.

 

[JoeDawg]

In fact paying attention to your experiences is a highly artificial and learned skill.

Well, sure. But that is not really what I'm referring to.

Animals and babies can't do it.

And this would be the example of 'experience'. Quite a lot of our adult life is constructed, but that's because we build ideas around 'sensations'. This is why differentiating between ideas and experience is important. Pleasure and pain, for instance, are immediate. They don't even need to be localized in time or space, within our minds, although quite often they are... and in that case part of the experience is constructed.

Yet still the very act of stopping and contemplating "redness" is a highly constructed

Contemplating yes, but experiencing no. Obviously if you are calling it red, you're attaching an idea to it. But there are lots of times, when we see something for the first time, we don't place it... within a framework, at least not immediately.

Impressionist painters are trying to simulate raw experience... on canvas. Not something I think you can really do successfully, but they try.

Yes, some things may be less mediated, less apparently constructed. But in the end, all experience is the result of some act of mediation, some constructive effort and not about naked witnessing.

Like I said, the fact biology 'constructs' a sensation in the mind is, I think, quite a different thing. Biology is an explanation. We as adults may reflect on sensations, almost immediately, and certainly our brains seem to want to categorize everything. But before we learn to do this, and on occasion when something intense or unexpected happens, we do have a kind of raw experience. And that is the sort of thing I am talking about. Adrenaline junkies crave this, and so do people who meditate.

A person's first orgasm, for instance, can completely shatter their reality. It's only after, when they organize thoughts around it, that it becomes 'constructed'.

 

[apeiron]

You are arguing here from personal prejudice rather than psychological or neurological fact. Which makes for an unproductive conversation as usual.

To ground your ideas, why not simply tell me at which point as a photon strikes a retinal receptor you feel that there is this supposed transition from raw input to mediated experience - constructed in the sense that the processing has begun in earnest.

Or if you prefer to focus on pain, then again, where after the finger is pricked with a pin does the percept swim into view. We know the neurology of the pain pathway. Where is the location where the magic of qualiahood achieved and there is an experience ready to be contemplated?

Yes, agreed there are degrees of mediation. And Sperling's iconic memory experiments would be a good line of evidence for you to be arguing here I would have thought.

But I fear you will never get the essential point that I am arguing. Which is that all experience is processing - mental construction - and ideas and impressions are then two extremes of this one process. They are not two different kinds of thing.

But if you insist on being dualist, taking the position that qualia are primal - naked conscious facts - then you will have to accept all the mystical and unscientific baggage with comes with such a philosophy.

 

[wofsy]

If experience is constructed then it must have been constructed from something. that something was either not experienced or it was non-constructed experience. If it is not constructed, then it is given.

Experience has two fundamentally different ingredients - that which is unchanging, such as the idea of space and that which is changing such as the perception of a color. One is certain the other unpredictable.

An attempt to isolate these two through e.g. through introspection does not invalidate the difference just because the attempt is contrived.

 

[wofsy]

Apeiron I don't know why you are getting insulting with me again. I feel like you are telling me that I am so stupid and biased that I don't have the right to be part of this discussion.

If that it true why not just ignore my posts?

I choose to be part of this discussion whether you like it or not.

I don't think you don't understand the basic fact of philosophy. That is that empirical theories of experience do not explain it they merely describe data that in the past has been detected in some experiment. You have no proof that these same outcomes will occur in the next experiment. These are merely constructs. The existential nature of experience - the true subject of philosophy (as opposed to science) - does not deal with empirical constructs. Your references to experiments and photons and whatever illustrate this that you do not agree with this. You think it is all mystical. The fact that you refute me by referring to neurological and psychological "fact" shows you believe this. A fact for you is some testable result of an experiment.

I think that you confuse philosophy with the theory of knowledge. Empiricists and positivists and others argue that statements about experience can only refer to testable results and can mean nothing more than the outcomes that they predict. This is a theory of meaning not a theory of the existential nature of experience.

If i say 'This is a piece of chalk', you say well what does that mean? What testable results does that imply? If I say experience is given, you say what experiments can I do that give that statement meaning?

I personally think that the idea that all meaning is really a collection of testable empirical outcomes is a definition that ignores the fundamental existential nature of experience. While it is valid for Science it begs the questions of Philosophy.

 

[Pythagorean]

to the OP (I haven't read this thread, I've participated in a couple like it):

Because we designed it to.

 

[apeiron]

Apeiron I don't know why you are getting insulting with me again. I feel like you are telling me that I am so stupid and biased that I don't have the right to be part of this discussion.

Ha, no I was insulting JoeDawg this time round . Sorry if that wasn't clear in the quoting.

The existential nature of experience - the true subject of philosophy (as opposed to science) - does not deal with empirical constructs.

I would be surprised if everyone agreed this was what philosophy was about. But maybe that is because my interests are clearly meta-physics and epistemology. Get these right and the rest follows I believe - even ethics and aesthetics.

As to my use of scientific facts, I follow Rosen's modelling relations approach. It is all about the interaction between ideas and impressions, models and measurements. So the "facts" inform the opinion, just as much as the opinon informs (or prejudices) the facts. You tend to see what you believe, and that is a meta-fact that our attempts to understand the world must deal with systematically.

This makes me impatient both with those who haven't worked sufficiently on forming their opinions (doing the philosophy) and noting the facts (doing the science).

I personally think that the idea that all meaning is really a collection of testable empirical outcomes is a definition that ignores the fundamental existential nature of experience. While it is valid for Science it begs the questions of Philosophy.

I was a psychology/biology student in the 1970s so felt all the frustrations of science's failure to tackle the issue of mind. And science is still generally failing to do its job here. It is the proper scientific (and mathematical) generalisation of the idea of mind which is my major life project. And I just don't see any real division between the philosophical and scientific aspects of this quest.

 

[JoeDawg]

But if you insist on being dualist, taking the position that qualia are primal...

Consciousness is primary to epistemology, ontology is something entirely different.

I never said anything about dualism.

You're nothing but a cranky troll.

 

[Math Is Hard]

In fact paying attention to your experiences is a highly artificial and learned skill. Animals and babies can't do it. And it takes a lot of practice and scaffolding for even modern Western adults.

That appears completely untrue. Even simple sea slugs can learn from experience. If they did not pay attention to an experience, it seems unlikely they would be able to pair it with an outcome and consistently modify their behavior (protectively) when a dangerous event repeated.

I suppose, they couldn't "mull it over" after the fact, but there was a sensory register (attention paying) at some point.

 

[apeiron]

That appears completely untrue. Even simple sea slugs can learn from experience. If they did not pay attention to an experience, it seems unlikely they would be able to pair it with an outcome and consistently modify their behavior (protectively) when a dangerous event repeated.

OK, this is getting ludicrous. Provide me with citations that Aplysia "pays attention" in Kandel's classic habituation experiments.

You in fact have this example exactly about front. Aplysia is genetically wired to respond to a prod and habituation is "learning" to ignore what is not actually dangerous. Or rather a simple tiring of the circuitry via the simplest feedback. No anticipation involved.

As to babies and chimps, I would refer you to Mead and Vygotsky. Feel free to debate the actual science.

 

[Math Is Hard]

Then I might be unclear on the definition of "attention". The way I see it, it can be something that is controlled:

http://www.mybrilliantkidz.com/wp-content/uploads/studying-child.jpg

Or it can be something that is uncontrolled:

 

[apeiron]

Controlled and uncontrolled would be a woolly distinction. And also irrelevant to my statement - "In fact paying attention to your experiences is a highly artificial and learned skill."

It is selectively attending to the "contents of awarenesss" - introspection - that I was talking about. Extrospection is what brains are designed for. Introspection is a skill humans cultivate (and is difficult because it is essentially unnatural).

 

[Math Is Hard]

Controlled and uncontrolled would be a woolly distinction. And also irrelevant to my statement - "In fact paying attention to your experiences is a highly artificial and learned skill."

It is selectively attending to the "contents of awarenesss" - introspection - that I was talking about. Extrospection is what brains are designed for. Introspection is a skill humans cultivate (and is difficult because it is essentially unnatural).

OK, I think I understand you better now. Thanks for clarifying.

So, it is your belief that we are not biologically wired for metacognition (thoughts about our thoughts), but that we can learn it, and that non-human animals are incapable of this?

 

[apeiron]

So, it is your belief that we are not biologically wired for metacognition (thoughts about our thoughts), but that we can learn it, and that non-human animals are incapable of this?

Correct.

 

[SixNein]

I’m reviewing my mathematics knowledge, except I’m looking for a different reason. I understand how it works you know, 1 plus 1 so on, I’m trying to understand why it works.

Pure and applied mathematicians and physicists have tied our understanding of reality with mathematics. They believe that if it computes it is real. This will do for a while, till we pose a question beyond our understanding.

But that’s for a different time. Why does it work?

Any takers?

Even though a calculation computes, the result may not reflect reality. To give you an example, let's assume a computer monitor has an area of 93.5 square inches. It's width is 2.5 inches larger then its length. What is the length and width of the computer monitor?

The area of the computer monitor is computed as A=LW.
We know the area so 93.5 = LW.
We also know the width is 2.5 inches larger then the length so 93.5 = x(x+2.5).
After we distribute the x, we arrive with 93.5 = x^2 + 2.5x.
After we set the equation to 0, we have 0 = x^2 + 2.5x - 93.5
Since we are too lazy to factor, we use the quadratic equation:

The b value is 2.5.
The a value is 1.
the c value is 93.5.

After we plug in the values and do some calculating, we arrive with two solutions.
The roots of the quadratic are X = 8.5 and x = -11.

So we test our results with the area formula A=LW.
The area a will be 93.5.
The length will be 8.5.
The Width will be (8.5+2.5) or 11.
We plug in the values 93.5 = (8.5)(8.5+2.5).
We do the addition (8.5+2.5) = (11) first.
Then we times 8.5 with 11 to get 93.5.
The equation now appears as 93.5 = 93.5, and it its a true equation.
Thus, our first solution of 8.5 works, and we can tell the length is 8.5 inches and the width is 11 inches.


The next solution was -11, and we do the same thing to test the results.
After filling in the values, 93.5 = (-11)(-11 + 2.5).
We again arrive with 93.5 = 93.5, which is a true equation; however, the length of the monitor would be -11 inches, and the width would be -8.5 inches.

Can a monitor have a negative length and width? As far as mathematics is concerned, the answer is yes; however, we are presented with a physical limitation. So we disregard the negative result in favor of the positive result.

 

[SixNein]

While I agree with your description of mathematics generally, I am not so sure that we can not have an ultimate mathematical/physical theory. Physicists differentiate between what they consider to be phenomenological theories and fundamental theories. For example the Shroedinger equation describes the spectrum of the hydrogen atom as a phenomenon but not in a fundamental way. this is because it takes coulomb forces as givens and does not explain them. But a theory like String theory attempts to explain everything fundamentally. Why could not a theory like this actually tell us everything exactly?

The problem is due to Godel's theorem of incompleteness. The theorem is very important because it shines a light on a fundamental limitation on systems. The limitation occurs when an attempt is made to explore properties of a system with the system. In a basic nutshell, the attempt cannot be complete and consistent at the same time. I personally think this manor of wording is very misleading to people, so allow me to reword it. In a basic nutshell, you cannot create enough axioms in order to have consistency and completeness. Since you do not have enough axioms, your system is incomplete. If an attempt to force completeness despite the lack of axioms is made, then the system will be inconsistent.

To illiterate the problem, I will create a very simple system.

Simple System:
In the United States, there is only one person named Joe who works as a professional landscaper. Joe mows lawns for a living. All inhabitants of the United States either mows their own lawn, or Joe to mows their lawn for them.

Limitation: If Joe does not mow his own lawn, then who does?

According to the system, if joe does not mow his own lawn, then Joe mows his own lawn.

See the problem?

This is why a TOE cannot be created. No matter how many axioms you add, you wind up with this same limitation. You can add axioms all day long with countless pages of complex details of the system, but you will eventually wind up with the Joe problem. If an attempt to force the Joe variable is made, the entire system becomes inconsistent.

 

[SixNein]

Ah, so you are a dualist ! I almost agree with what you're saying here, except for one minor point.

You say that the reality of ideas is indisputable. I agree. But for real ideas to be discovered as opposed to being invented or constructed, they must be real before they are ideas in the minds of any particular mathematician. This leads you to Berkeley's argument that for these ideas to have timeless existence, they must be existing in the mind of God.

If ideas are real, then either they are created in the minds of those who are thinking them, or they exist permanently and objectively in an eternal mind. In order for real ideas to be discovered, there must be an eternal mind in which they exist prior to their being discovered by our minds. Also, in order for ideas to be objective, they must exist in an omniscient objective mind.

If math is ideas that are discovered, then there exists an objective and eternal mind.

If math is relations and not ideas, then we don't have this problem. Analytic relations may be discovered, but they are not objects and do not have such a thing as existence. That is what I mean by the idea that they are purely formal. Their method of discovery is deductive and logical rather than scientific.

People should drop the human element from the entire discussion because the human element complicates the problem. I think it would be best to form the question outside of the human mind completely; instead, people should assign the question to a computer. If a computer finds the solution to p=np, did the computer discover the solution or invent it?

 

[Mattara]

The problem is due to Godel's theorem of incompleteness. The theorem is very important because it shines a light on a fundamental limitation on systems. The limitation occurs when an attempt is made to explore properties of a system with the system. In a basic nutshell, the attempt cannot be complete and consistent at the same time. I personally think this manor of wording is very misleading to people, so allow me to reword it. In a basic nutshell, you cannot create enough axioms in order to have consistency and completeness. Since you do not have enough axioms, your system is incomplete. If an attempt to force completeness despite the lack of axioms is made, then the system will be inconsistent.

For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.

There are places where this incompleteness theory is not applicable.

 

[wofsy]

The problem is due to Godel's theorem of incompleteness. The theorem is very important because it shines a light on a fundamental limitation on systems. The limitation occurs when an attempt is made to explore properties of a system with the system. In a basic nutshell, the attempt cannot be complete and consistent at the same time. I personally think this manor of wording is very misleading to people, so allow me to reword it. In a basic nutshell, you cannot create enough axioms in order to have consistency and completeness. Since you do not have enough axioms, your system is incomplete. If an attempt to force completeness despite the lack of axioms is made, then the system will be inconsistent.

To illiterate the problem, I will create a very simple system.

Simple System:
In the United States, there is only one person named Joe who works as a professional landscaper. Joe mows lawns for a living. All inhabitants of the United States either mows their own lawn, or Joe to mows their lawn for them.

Limitation: If Joe does not mow his own lawn, then who does?

According to the system, if joe does not mow his own lawn, then Joe mows his own lawn.

See the problem?

This is why a TOE cannot be created. No matter how many axioms you add, you wind up with this same limitation. You can add axioms all day long with countless pages of complex details of the system, but you will eventually wind up with the Joe problem. If an attempt to force the Joe variable is made, the entire system becomes inconsistent.

I don't see how Godel's theorem applies here. I thought it applied to specific models of axiomatic systems.

 

[SixNein]

I don't see how Godel's theorem applies here. I thought it applied to specific models of axiomatic systems.

Formal, Finite, and Self Referencing are the requirements.

Are physical theories formal? Yes.
Are physical theories Finite? Yes.
Are physical theories self referencing? Yes.

*poof*

 

[vectorcube]

Last time i was in this forum, there was this stupid guy that say "the laws of nature is derived from logic". Obviously, the guy is misguided, but i often feel that common people don ` t see the difference between math and physics. Math is a tool used to describe physical laws, and the tool to tease out the consequence of the laws.

 

[wofsy]

Last time i was in this forum, there was this stupid guy that say "the laws of nature is derived from logic". Obviously, the guy is misguided, but i often feel that common people don ` t see the difference between math and physics. Math is a tool used to describe physical laws, and the tool to tease out the consequence of the laws.

Physical laws are mathematical in my opinion. So math is not a tool only. It is the law.