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# Archive for the ‘Mathematics’ Category

## Examples of Math being Empirically Justified

Posted by Tanas Gjorgoski on June 25, 2008

Richard at Philosophy Sucks! gave this example of how math truths might be empirically justified:

Suppose that you had two pens of sheep; one with 6 and one with 7 sheep. Now suppose that you counted the sheep individually in each pen (and got 6 and 7) and then you counted all of the sheep and got 14. Suppose you did it again. 1. 2. 3. 4. 5. 6. Yep six sheep in that pen. 1. 2. 3. 4. 5. 6. 7. Yep seven sheep in that pen. Then all the sheep. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Suppose that this was repeated by all of your friends with the same results. Suppose that it was on the news and tested scientifically and confirmed. Suppose that this phenomenon was wide spread, observable, and repeatable. If this were the case we would be forced to admit that 7+6=14 is true therefore mathematics is empirically justified.

I got to say I’m little disappointed with this example, because it doesn’t work at all… In the comment there I said that to make things fair for the rationalist, we should after counting the sheep from first pen, do both those things in same time: a)continue counting the sheep in both pens and b)count the sheep from the second pen.

So, we count the sheep from the first pen, 1. 2. 3. 4. 5. 6, and then for each sheep in the second pen we continue both countings – – so (7. 1.) , (8. 2.) , (9. 3.), (10. 4.), (11. 5.), (12. 6.), (13. 7).

What we did is then, that we counted all the sheep (13), and counted the number of sheep in both pen (6 and 7).

But anyway, I still have trouble figuring out what would it mean for math claims to be empirically justified. It is not like as if we can find mathematical entities in the world, so that we can test them. We could do this kind of counting and be surprised that everytime when we get 6 and 7 we get 13, but surely it is weird thing to do – given the we agree of how we count, it can’t be otherwise!

Posted in Mathematics, Philosophy | 13 Comments »

## Pythagoras A Posteriori Style

Posted by Tanas Gjorgoski on May 21, 2008

For all you scientismists and psychologismists and anti-apriorists out there

He says “Proof of the Pythagorean theorem”. So, if you weren’t sure…

## The Meaning of Few Different Words Within the Illusionary World

Posted by Tanas Gjorgoski on May 19, 2008

For the few past posts I was thinking/writing on the issue of how I could be a brain in a vat, to which random electrical impulses are fed, but so it happens that by mere chance, I’m under illusion of living a normal life in society.

Keeping inline with my externalistic preferences, I said that while in the vat the subject can’t become aware of anything real, she can become aware of different possibilities. She will become aware of possibilities of objects, multitudes of objects, multitude of objects sharing some similarity (kinds), possibility of other subjects perceiving, possibility for open possibilities in the world, related to this acting and practices, related to this possibility of language as practice, and so on.

I want here to add few thoughts on the words used to describe the scenario, like ‘brain’, ‘vat’, ‘electrical’, ‘impulses’, ‘chance’, ‘life’, ‘society’, ‘illusion’, and so on. I want to comment on the issue if those words could have same meaning unrelated to the fact if I am a subject of perpetual illusion or not.

‘Vat’ seems pretty unproblematic. A large container used for storing or holding liquids. Vats are surely not natural kinds, nor is having idea of one dependent on there being one. I guess in general for artifacts we could say this… for one to create (engineer) something new, one needs to be able to think about the possibility of such thing, even before there are such things. The issue is though, how abstract those possibilities are. On one side we could have abstract things like for example Turing machine, with abstract algorithms (e.g. Quick Sort), but maybe the idea of vat (the awareness that there could be vats) is little more problematic, as it depends on the idea of liquids. I don’t know… I don’t see liquidity as problematic either, it seems to me that it describes a possible property of a substance, and that we could distinguish the liquidity as property from the reason for liquidity. And liquidity as a property there is related more with what how the substance behaves, and as such we can become aware of possibility of such behavior.

Brains… What do we mean by ‘brains’? Generally, we tend to find this organ in humans and other animals higher animals’ heads. So, I think we need to put attention to ‘humans’, ‘animals’, ‘organ’ and ‘head’. ‘Humans’ in one sense are a specific natural kind, and as I said, I’m inclined to think that our words within the vat that was supposed to mean natural kinds can’t refer to real natural kinds (as they weren’t based to real multitudes – so similarly to how proper names can’t mean real things, as they weren’t based on those real things). But, from another sense, ‘humans’ might be taken to mean – the species to which I belong. And the idea of ‘species’ along with the idea of ‘animals’ and ‘kinds of animals’ seems much more abstract that it would refer to the same thing, be I under perpetual illusion or not. Of course, it might be also that for the case we have here, we don’t need to go as far to other animals, and kinds of animals, but just to think of the kind to which I as a subject belong (defined thought the possibilities of becoming aware of all those different things, and possibilities for acting), and further the idea of having body, and having head, and having something in the head which is related to being a subject.

‘Electrical’ seems very problematic, as it refers to a specific natural phenomenon, which isn’t much a specific property of the things, but something that we figured out through science. I don’t know though… The physical laws take very abstract form, and the notions which are related to the physical theories (like atoms etc…) are also kind of abstract.

The possibility of other subjects is i think non-problematical, and the possibilities for those subjects to act in different ways when together, including possibility of communication, different social relations and so on – I think if one becomes aware of those as possibilities within the vat, and if thinks of those within the vat, those are the same things of which we may think of. I think it is similar to the case with engineering I described before. The communication, or different ways of acting towards other subjects are I think possibilities of which one might become aware even before those ways of acting towards others exist. For sure, we might be inclined biologically to take some of those ways, but we also think and invent new ways of how to relate to others, how to solve problems in our relations, how to better do different things, and so on. And if some person becomes aware of some of those possibilities within the vat, I think he can then, when outside of the vat share the same ideas with others (real others).

I guess it is much more important what the scenario meant to point to, and that is a certain possibility which is more abstract than the words that were used to describe the scenario. And the possibility is that I as a being which can think, perceive and so on, can be in fact subject of perpetual illusion. And ‘illusion’ is I think less problematic in this sense. As I described in some past post, it is about possibility that the subject can’t distinguish between two different experiences in which he takes part. And I think the brain in a vat which is under perpetual illusion, and us, when thinking of illusions are thinking of the same thing.

Anyway, after I noted in last post that there might be some problems in the details, I thought it would be interesting to do some analysis, so… that’s about it. Probably, if nobody objects, I will have another post (or two) about the perpetual illusion scenario.

## Doing Metaphysics In A Vat

Posted by Tanas Gjorgoski on May 18, 2008

Starting with the scenario from previous posts…

A baby’s brain is put in a vat, and connected with wires to a generator of random electrical impulses. By mere chance, though random, those electrical impulses happen to be such that the subject which is related to the brain is under an illusion of living what we would call normal life within a society. (If you think that other parts of the body are needed for such an illusion to be possible, just imagine the whole body in the vat, like in Matrix)…

I don’t see anything contradictory with the possibility that I, myself am subject related to such brain (or body) in a vat.

Accepting this means that I accept that whatever I said, has same meaning, unrelated to the fact if I am living real life or pure-chance-random-generators life. And that I can think of this issue, unrelated to the fact if I am subject of illusion or not. (Though of course, there are some problems with the details here. First there might not be any electrical force in the real world, or there might be no such things as brains, but I think that what is important that the general idea of being a subject of perpetual illusion is the same thing be I subject of perpetual illusion or not.)

As one, if I am subject of illusion the things I’ve seen are not real. So, the people, objects, animals, etc… were not real. But for me to be able to think and say this, I need to have a valid notion of ‘real thing’, to distinguish those from ‘illusions’, and so on. The problem is of course, if nothing I’ve seen is real, how can I have notions like those.

In the past post, I presented the idea, that while I never became aware of anything real while subject of perpetual hallucination, I’ve become aware of possibilities of different things. It is interesting, relating to the issue of what kind of notions I need to have in order to be able to say that I might have been subject of perpetual illusion, to think about what kinds of possibilities I might be aware… So, here is some list of important notions, I think I am having, unrelated to the issue if I was subject of perpetual illusion or not…

I must have had become aware of the possibility of objects, multitudes of things, kinds of things, perceiving things, others perceiving things (by ‘others’, I don’t mean necessarily people, at least not in biological sense, as there might not be such things as people), possibility of open possibilities in the world and acting in order to actualize a possibility, possibility of communication as a way of acting (in different ways – like to communicate what one knows, to ask, to order, and so on), possibility of all kind of different social relations.

We can contrast those things with a)I have simply been wrong in using name to refer to the illusionary thing, as there wasn’t anything there to which I was referring – I was simply wrong. It was similar to fictional things, and as such as much something in real life is alike that illusionary thing, I can’t say that I meant that real thing, by using the name. b)Also, to some amount when talking about multitudes of illusionary things sharing some similarity, as much the similarity can be seen as possibility, it is questionable how much to this notion it is critical that the multitude is seen as a real multitude, about which we can figure out more truths. So, I would be inclined that in lot of cases to say that if we referred to illusionary kinds, those words can’t mean real kinds, like lemons, water, and so on…

Given this contrast, I want to propose this principle (let us call it… metaphysics while in illusion principle, or maybe better just P1):

P1: We can think a priori about (and only about) things of which we can think of, even if we were subjects of the perpetual illusion scenario.

This will include notions like object, multitude of objects, kinds, possibilities and acting, society, practices, communication, numbers, change (also time and space), good and evil, and all those other notions, that a subject could think about even living life under perpetual illusion, and which are the same things that we (living real lives) are thinking of.

So, what would be the argument for P1. Let me attempt to give an argument…

Given that I can think of something while subject of perpetual illusion that other people which live real lives are thinking about, there is nothing which would determine the reference to that thing which is not already there, when I have the thing in my mind

Given, that we have this thing (this notion) clearly before our mind, we can then try to figure out possibilities and necessities within or related to that notion.

OK, that was I guess kind of a bad argument (in the sense that I don’t think it is very convincing), but besides the argument let me also point to one case of notions on which lot of people might agree with P1. That is the case of mathematics… While I would think it is pretty straightforward that science done under illusion can’t be real science or tell us anything about the world, the case with mathematics seems different. I can think about triangles, angles, sums, and so on while in ‘hallucination-world’, about the same triangles people think in the real world. But also if I become aware of the proof of Pythagorean theorem, because some illusionary guy on illusionary place seemed to teach it to me, it won’t make the knowledge I gain in such way wrong. What is important that I understood and comprehended the Pythagorean theorem (even as it happened – as a result of very peculiar events). So, while specific case, I think this can nicely pump your intuition into accepting P1.

## Resolving The Mind-Body Issue, Part 4

Posted by Tanas Gjorgoski on April 29, 2008

This is the last part of the series of posts about how I think we can make sense of the mind-body issue. The idea is to provide alternative to different kinds of physicalism and dualism.

The story so far:
So far, we took those steps:
1.Returning the content which was previously pushed under the “rug of the mental” back into the world. What is left to the mind are the abilities to perceive, imagine, plan, etc…
2.Moving away from atomism/constructivism towards seeing the world in terms of what is actually going on, and aspects of what is actually going on, and taking both which appears in the physical picture and what we returned into the world as aspects of the world.

And in the third part, I didn’t advance the idea further, but pointed to a problem in the idea… For something to be an aspect of X, it means that the facts about X are based on
a)what is actually going on, and
b)the nature of the aspect.
For example, the facts about the contour of the face seen from certain side (as an aspect), will depend on a)the three-dimensional shape of the face and b)the angle from which we are seeing the face.

But what we are seeing in physical nature are that there are physical laws, which can be nicely put in determinate mathematical equations. If we take those to be facts about the physical, it surely doesn’t seem that they are aspect of anything else – they seem self-contained and independent. Sure, there is the quantum indeterminacy, but that one is nicely isolated. When we have aspects, we expect that the facts about the aspect to be in more “organic” dependency on the whole.

What we do next:
So, we get here to what I think is third part of the solution of the mind-body issue. In short it is this… The physical laws are metaphysically necessary relations between different aspects which we see as being physical. The form of those laws is the conditional: whenever p is true about something, also q will be true about the same thing.

Let me explain this through an analogy with a case that can be understood more easily. Imagine that we ignore lot of things of whatever is going on, and put attention just on the geometrical and arithmetical aspects of whatever is going on. It so happens, that even everything is changing, in certain cases we can safely ignore the changes and analyze the non-changing aspect of the situations as if there is no changes occurring.

We can say about those cases that we are safe to apply a geometrical or arithmetical notion to the situation. With that, we get to the antecedent (p is true about X) of the conditional. For example it might be ‘the base of the house is square, with sides 5meters and 4meters’. From there it is possible to apply the mathematical truth ‘whenever something is square with sides a and b, it will have area a*b‘ , so we also have the consequent (q is true about X) – The area of the house is 20 square meters.

The point is that if for pragmatic reasons we can safely ignore everything but geometrical and arithmetical aspect of whatever is going on, it is normal that whatever mathematically necessary relations hold, will hold for this aspect. We can see how also, because what we took is merely an aspect, it might happen at any time that the antecedent of the mathematical necessity becomes invalid for reasons which are not captured by that aspect. So, we get to a situation where the mathematical aspect has 1)mathematically necessary relations which hold, but 2)the mathematical notions in some situations will not be applicable, for the reason that the situation goes beyond this aspect.

I think it is now clear, what is the proposed explanation of the seemingly self-subsistent regularities of the physical aspect. I said before that the facts about aspect will be dependent on two things: a)what is actually going on and b)the nature of the aspect. The solution is then, to connect those regularities known as physical laws to the later – to the nature of the aspect itself, and NOT to whatever is going on.

There is lot more to be said on this, and I’ve discussed this issue several times, but let me just add few brief note to this… The nature of the physical aspect is defined by the nature of the measurement of movement through measurements of space and time, and further to the way other physical properties like force, inertial mass, energy and so on, are related to those. Further, I think that it is in the nature of those measurements (or in their concept) to have different symmetries, and it is that which I think is a base for the metaphysical necessity of those laws. However, that this is just an aspect is shown in the cases of quantum mechanical collapses, to which the physical laws can’t apply for the basic reason that whatever is going on goes beyond one of its aspects – in this case physical aspect. In those cases, we can’t apply the notion in the question to the situation, similarly to how we sometimes can’t apply mathematical notions to the situation and can’t analyze the situation in terms of mathematical necessities.

So, this would be the last part of the solution for the mind-body issue. As I said at the start of the post, it does go far from the prevalent paradigm of the times, but given that the mind-body issue surely seems as an impossible problem to solve in the paradigm of our times, it should be clear that there is something wrong with that paradigm.

## What Would It Mean for Math To Be Empirical?

Posted by Tanas Gjorgoski on October 13, 2007

Over at Obscure and Confused Ideas, Greg discusses possibilities for arithmetic to be empirical.

But what would it mean for arithmetic (or geometry) to be empirical?

I proposed this distinction:

a)Mathematical description can be applied to a given system. But we empirically find out that the mathematical truths which follow from the description are not true in that system. The example would be that we can describe something as ‘two’ (i.e. the system IS system of two things), but through empirical research we find out that there are no “one and one more thing” (of course while still there are two things).

b)Mathematical description can’t be applied to a given system. For example we can’t use simple arithmetics to track truths about number of rabbits in some room, if we just add one to a sum for a rabbit that enters the room, and subtract one if a rabbit goes out of it. That because rabbits can be born and die within the room too. However this doesn’t mean that arithmetic is wrong, or that it is not true that 1+1=2.

I think only in cases like (a) we can speak of arithmetic or geometry to be empirical.

But as a commenter Drake there pointed (btw, check out his cool MySpace page. But wait, first finish reading this post!):

The criterion for a state of affairs S’ falling under a mathematical description D is that D “holds” in all respects relevant to S. If it doesn’t, some other description D* is required. Conversely, to see that
D is not the right description for S, we have to see that D in some relevant respect doesn’t hold.

So, it is not clear that cases like (a) are intelligible at all. And I agree. To relate to the example – what would it mean that there are two things, if there are no one and one more thing?

BTW, I don’t use 1+1=2, as I think the formalism might hide the basic analytic truth that whenever we have two, we have one and one more thing, or the other way around. This is not saying that “two” means nothing but “one and one more thing”; but that when we have two things we can both think of them as a pair (“two”), or think of them as separate (“one and one more”). But as this is true just in virtue of the meanings of the terms, it is analytical truth.

Also, further interesting note would be, if we e.g. find something like (a) (yes, I’m saying it is unintelligible, but just for sake of argument let’s say it is) , e.g. a system where we have two things, but we have one and one more and still one more thing, would that mean that we should now change our math books, and put in there that 2=1+1+1?  What we will do with systems where we have two things we have one and one more thing? Will this show that arithmetic is inconsistent or something? I think, again, what we would have is a case of (b).

Two further cases that might be interesting:

A. Theory of Relativity

Can we say that the theory of relativity is case of (a)? I think not. What it shows, I think, is just that Euclidean geometry is not applicable to things in movement/which apply forces (gravity) to each other. However, what is put in place of Euclidean geometry is not a geometry that we have “constructed” from empirical research. It is as a priori (and non empirical) as Euclidean geometry is. What was empirical was the figuring out which of those mathematical descriptions fit the universe.

B. Unavailable permutations of particles in QM

Both ‘classical’ and ‘quantal’ objects of the same kind (e.g. electrons) can be regarded as indistinguishable in the sense of possessing the same intrinsic properties, such as rest mass, charge,
spin etc…That a permutation of the particles is counted as giving a different arrangement in classical statistical mechanics implies that, although they are indistinguishable, such particles can be regarded as
individuals…If such permutations are not counted in quantum statistics, it follows that quantal particles cannot be regarded as individuals … In other words, quantal objects are very different from most everyday objects in that they are ‘non-individuals’ in some sense. (SEP)

Is this the case of (a), do we have here a case where we have two particles, but not one and one more? This might be closest something can come to (a), but one can ask having in mind Drake’s comment, why do we think that there are two things after all? If we don’t think that there is one and one more thing, shouldn’t the conclusion be that there are not two particles either? I’m inclined to this second conclusion.

Anyway, any further idea of what would it mean for mathematic to be empirical?

Posted in Mathematics, Philosophy, Physics | 9 Comments »

## Is Square Root of 2 Infinite?

Posted by Tanas Gjorgoski on September 16, 2007

Vallicella at Maverick Philosopher has little discussion of sqrt(2) [square root of 2] in this post. It is related to his previous posts on the issue of actual vs. potential infinites. In the post Vallicella says:

Suppose you have a right triangle. If two of the sides are one unit in length, then, by the theorem of Pythagoras, the length of the hypotenuse is the sqr rt of 2 = 1.14159. . . . And yet the length of the hypotenuse is perfectly definite, perfectly determinate. If the points in the line segment that constitutes the hypotenuse did not form an infinite set, then how could the length of the hypotenuse be perfectly definite?

What I want to point is that the hypotenuse doesn’t have any particular number related to it. It is square root of 2, only in relation with the sides which *are taken to be* 1. The nature of the number to be a ratio is shown here… the number which we assign to the hypotenuse is about the ratio between it and the other sides. However, we could instead take the hypotenuse to be 1, and the sides would be then sqrt(1/2). Or we could take the sides to be 3, and the hypotenuse would be then 3sqrt(2), etc… We see that what doesn’t change is the ratio, and what is perfectly definite here is the ratio.

Now, is there anything infinite here? I don’t think so… the hypotenuse and the other sides of the right triangle have a certain and limited length. Also, such as they are, we can speak of the ratio they form, and hence speak of numbers in relation to them. But numbers are not merely sums – we might *try* to express the ratio of the hypotenuse and the side of the right triangle through a sum, but in the given case we will fail. And this failing, this impossibility to finish with the “translation” of this ratio from its “definiteness” into our numerical notation, IS what we have as infinity here. We just can’t succeed into finishing this translation *because it can’t be done*! Some things can’t be translated into aggregates or sums. That is because there is more to this world than aggregates (or sums).

This ratio also has relations to other ratios which can be expressed as ratio of whole numbers. It can be bigger than some of those ratios and smaller than others. And just because of it, there is possibility of getting as close as needed to this ratio by using ratios of whole numbers. (Of course the ratio of rational numbers also falls under ratio of whole numbers as a/b:c/d is nothing other but the a*d/b*c ratio of whole numbers).

Of course, we can continue this translation as much as we can, and only because it will never succeed, we might be inclined to say that the product of this translation is infinite. But if we search for an ‘actual infinity’, it will not show itself in the results of some repeating iteration, but what is just “imperfectly” expressed by this sum, is as actual, definite and finished in the form of the term sqrt(2) (*and all of different cases which share the ratio with the ratio between the hypotenuse of the right triangle, and it’s side*, or which can be showed as identical).

Seems to me, that if we want to talk about actual infinity, we can just talk about being infinite in relation to something else – i.e. that that former can’t be reduced to the later.

Posted in Mathematics, Metaphysics, Philosophy | 7 Comments »

## More Thoughts On Mind/Body Issue

Posted by Tanas Gjorgoski on April 28, 2007

I wrote several times about my thoughts on mind/body issue. I will try to write one post in which I would clearly explain the full picture of how I see this issue.

Physical World As Abstraction

To repeat in short, I think that through physics we put our attention on abstraction – i.e. on specific things in the world, while ignoring others. That is because it is those things which are approachable through scientific (physical) analysis. In such way, we put attention on different quantifiable abstractions like position, moment, energy, velocity, energy, frequency etc…, but we ignore others which are present in the world and which we are aware of, like emotions, beauty, music, colors (or some other things usually put under ‘qualia’) and the awareness of things and/or possibilities open in the world itself.
By doing that, we are left in physics with an impoverished and abstract world. Maybe a good metaphor would be like when in 3d rendering software you turn off the shading, and are left with just the wire model.

What is further figured out about that impoverished world is that there are laws which hold among those abstractions (quantified this or that). Those show up as a necessary relations which hold among specific abstractions of the system, if the system as a whole falls under certain abstraction.
Now, it is true that the method of empirical sciences is such that we can’t ever be sure that we figured out that there are such necessary laws, but I think that all things for now point that the nature of this impoverished world is such that this kind of necessary relations among abstractions do hold, if not the “physical laws” that physics got to today, then some other.

So, I believe that the physical world as an abstraction of the world, and that we shouldn’t equate the world with the physical world and than try to “pull out” what has been left out (intentionality, colors, sounds, ethics, beauty, etc..) from the realm of the mental as physicalism and dualism try to do.

Metaphysical Theorems

Anyway, this kind of view goes nicely with what I’ve been also writing about here, and that is that the physical laws can be known a priori and that they are similar to the mathematical theorems, just that they include not just quantity, but more metaphysical concepts. Let me present a picture of how these two views – a)that the physical world is abstraction from the world and b)that the physical laws can be known a priori (or that they are metaphysically necessary) nicely go together by making an analogy with world – ‘math world’ relation.

I take it that math ‘laws’ can be known a priori. For example you don’t have to get to empirical research and measure the sides of right triangles, in other to postulate and then show through further measurements that the Pythagorean ‘law’ holds between the sides of right triangles. Instead, through mere knowledge of the concepts and their meaning, and through (little?) thinking one can figure out the relation, and come to know a priori the relation that we call Pythagorean theorem. (Some would like to include the e.g. curvature of space-time as an argument that we can’t know that Pythagorean theorem holds a priori, but that just means that if the system doesn’t fall under that certain abstraction – i.e. a right triangle in flat space, then the necessary relations between the sides won’t hold. After all, in curved space time right triangles, we have another theorems that do hold.)

Let’s now return to our “math world”. It is clear that in the world in which we live, we can put our attention to one thing while ignoring other things. We can in such way ignore the specific types of things, and only speak about their quantities as they appear to us. Or we can ignore in what way we determine some position, distance, etc…, and think of the world merely in math (arithmetic and geometric) terms. Having impoverished the world in such way, the situation in the world falls under (or is) the abstract situation that we work with in math. And for this abstractions, as long they fall under (or are) the abstract concepts of math, the math truths will hold. So, to say, as long from the situation in the world we can abstract a right triangle, the Pythagorean theorem will hold for whatever is related to the sides of that triangle (e.g. if we abstract right triangle, from the centers of three balls, the Pythagorean theorem will hold between the distances among the balls). Now, we get into situation, similarly as the one described with the physical laws, where there are specific necessary relations between certain abstractions if the system/situation falls under certain other abstraction. And again those necessary relations hold among things in the world, as we didn’t abstract them from anything else but from the world.

So, when we talk about the relation between the world and impoverished ‘math world’, we can say that if we abstract certain things from the world, we will end-up with a world which is fully ruled by the math laws. (Again, as far the thing falls under the given abstraction, or as far the thing is the abstraction.)
So, what I believe is that the situation is same with the physical laws, that is with those (supposed) necessary relations that hold between the abstractions that physics puts attention on. I take it that those necessary relations, would be in such way something like “metaphysical theorems”, that describe necessary relations not just between dimensionless quantities, but also of different physical concepts like time, space, energy, mass, and so on…

This would be similar to Kant’s view that the physical laws are a priori, just that in this case as the abstractions are from the world, the physical laws are about the world (as a real, and not merely phenomenal world distinguished from the noumenal world). And while the Kant took the absolute space and time as given as absolute, following Einstein (and Hegel for that matter), we can look at those merely as abstractions. The view that there is such metaphysical theorems (which I would think would be the main task of metaphysics to get to) might seem very optimistic, but let me point that lot of the reasoning in theory of relativity is a priori, and how the symmetries, which I take to be likely metaphysically deductible for lot of things (e.g. the symmetry of space or time) are one of the main principles of modern physics.

Doesn’t Everything In The World Happen According To Physical Laws?

Now, let’s assume that metaphysics can in fact, get through a priori reasoning to its end – i.e. to metaphysical theorems, that will be necessary relations that will hold for given abstractions when some part of the world falls (or is) under certain abstraction. Having done this, of course, metaphysics would have finished what the physics is after – the theory of everything, and it would have also shown that whatever is nomological and metaphysical modalities coincide.
But, where would that leave us, what would it mean?
One of the things that would mean, is that we need to change our view of physical laws as things which “control” the development of the universe, to a view of them as necessary relations between certain abstractions as far as something in the world falls under (or is) that abstraction.
Same as there is no “math laws” which control the universe and make sure that when we add one thing to another we get two things. And same when we have three points in a situation that describe right triangle, there isn’t some law that makes sure that the relation between distances satisfies the Pythagorean theorem.
So, to reinterpret that, in such case (if the metaphysics presents us with those metaphysical theorems, formerly known as physical laws), we could say that for any system in the world, its behavior through time, as far as the system falls under some abstract description, will necessarily satisfy those metaphysical theorems. But, and this is the interesting possibility which connects to the start of this post, the world doesn’t have to nor is the impoverished world in which only physical concepts are left. So, the development of the world as far as it can’t be described merely by physical concepts, won’t be fully determined by physical concepts. Same as the world in which things disappear can’t be fully determined by the math concepts – the system in which things appear and disappear, just can’t fall under abstraction of (or isn’t) a simple quantity.

So, where would those metaphysical theorems (formerly known as physical laws) not hold? An obvious answer is – in the situations in the world that include things which are ignored by the physical analysis. And we mentioned which are there – the main one I think is intentionality, our awareness of things and the things that fall there (and dependent on this), like colors, sounds. Also things like emotions, art, morality and so on.

What If The Physical World Is Just An Abstraction (Aspect)

So, if the world isn’t merely a physical world, and if the physical laws (metaphysical theorems) hold just as far as a specific part of the world falls (or is) a physical world, what would happen if we try to analyze the world in the physical terms in the situations that in fact include things which are ignored by physical (while present in the world). So, things like intentionality, colors, sounds, and so on…
First, let me say, that according to this picture, when we analyze some such a situation in terms of physical, for example the situation of me seeing a rabbit, we should be analyzing the situation as including both me and the rabbit. It is this whole situation where I see a rabbit, and which includes elements (for example intentionality) which are ignored by the physical picture, but there is no sense in searching for some correspondence in the physical picture by further limiting the analysis of the situation just to whatever is going on in my head (in my brain).
So, by this picture, even physical picture will always be impoverished, it makes more sense to analyze the full situation which includes the world, the body and the brain, and not just the brain.

Anyway, what I think is important is that if the world is not merely physical world, and because we are aware of it (after all, that’s why I’m writing this post), that things happen in the world not just because some relation among abstractions should be necessary, but for reasons which are connected to things in the world which are ignored by the abstractions of physics. In this picture, however this doesn’t go against the necessity of the physical laws, because the physical laws hold only as far the part of the world falls under some abstraction. In another post (Physics vs. Physicalism) I was analyzing more specifically how the QM indeterminacy can be related to this, so that the reasons why the collapse is such and not other way, can be explained by this: the world is not merely a physical world, and even in the necessary relations there will appear things in which the reasons from the “richer” world will be mapped to the functioning of the impoverished world.

## Lifeless Laws

Posted by Tanas Gjorgoski on December 16, 2006

Physical laws present to us, (or would present to us in ideal case), the necessary relation (necessary for any measured system in this world) between different values we measure on a system.
Number is ratio, and measurement is applying of such ratio to certain quality of the system by comparing to the quality which is taken as unit. In such way through measurement, we abstract number from the physical system.
Time is merely one of those values which are measured.

Can we say then that physical laws are not the thing that give life to universe? They just say that for any system in the universe, the values will fall into certain relation.

## Familiar Faces, Gestalts and A Priori Truths

Posted by Tanas Gjorgoski on December 10, 2006

We easily recognize a familiar face in a group of people.
We recognize her face in a moment, without analyzing the details of the face. In fact, if someone has asked us to describe features of that face, or to answer questions about it while not looking at it, there is a good chance that most of us would fail to do so. I’m thinking here about features like color of the eyes, the form of the lips, the length of nose and so on.

I would say that as far as we can answer such question or provide descriptions, it is because either we can bring that familiar face from the memory before our “inner eye”, or because we had explicitly attended to that feature in the past (we might have had intentionally focused on it, or it attracted our attention).

When we see a person’s face, we don’t see it as an aggregate of multiple features, but we see it and later recognize it as a whole, as a gestalt. But while we can remember and recognize things as a whole without learning about specific features, still in any case when the face is before us, it is there as analyzable. When we look at it, the features are there – open to our possibility to focus on them, open to our skills for analysis which may be “triggered” either spontaneously or by some kind of reflex (e.g. when something attracts our attention).
And in those cases we can attend to different things, we can attend to one eye, or to the other eye, or to both eyes at once, we can attend to something which we previously didn’t notice – for example we can attend to a small part of the curve of the left eye, or to the relative brightness of a specific place vs. another, or to the number of speckles on the right cheek, etc…
But while we attend to one of the things, our consciousness is not limited to it. As when at the start the gestalt was there as gestalt presenting the possibility for wealth of different abstractions; now when we attend to a specific abstraction (feature), it is there as an abstraction – as a part of the whole which is still there to return to.

So, let me now turn from familiar faces and gestalts to a priori truths.
In a previous post I was wondering if some tests in developmental psychology might be taken as a hint that the infants believe that 1+1=2 based on intuition. While it might be taken as a hint, as it was pointed in the comments by Pete and Curtis those experimental results hardly show anything decisive – one can explain the results in different ways which wouldn’t include any mention of intuitive truths.
But let me look at the issue in the context of the previous discussion…
When we have in front of us two things, we can see them as a gestalt, as “two things”, something which we can see and recognize. But the same gestalt, which we can name as “two”, is also analyzable – it is implicit in that gestalt that I can attend to the one or to the other of those two things. And while attending to one of them, as in the case with the face, the other one is not disappearing – it is still there. And I can spontaneously change the way I attend to the whole, I can focus on the other one. So, in that whole of two things, I can switch my attention, look at the both things as a whole (two), or I can attend to each of them separately.
In this way, we are presented with the a situation, in which the same gestalt is analyzable either as one two, or as two ones. And that goes for any gestalt which I can imagine – as long the gestalt is two, it will present possibility for attending to two separate ones (among which we can switch the attention).
This a priori relation of possibilities is there, be it if what we characterize as two is in front of us, or if as in the case with the experiment done with the kids mentioned in the other post, one or both of the things are tracked (as hidden behind the screen), or even if we have imaginary gestalt.
This is, I think, what is behind our intuitive understanding of what we express by 1+1=2. The equation shouldn’t be taken as identifying two separate sides, namely a)1+1 and b)2 , but as expressing that the whole, if it is characterized as two, can be also characterized as one and another one. The identity is in the whole, and the equality is expressing the necessity that in every possible world the whole which is 2, is also 1+1 and vice versa.

And at the end, let me finish with a doubt I have…
While I’m pretty convinced that I have grasp of the a priori truth of the equation that 1+1=2 (meaning what has been just described), I’m not very sure what to say about the issue if the truth of 1+1=2 is analytical.
On one side, it seems to me that the potential to analyze the whole into separate things (i.e. to focus on the one and the other) is what is required for something to be named as two, but on other side having on mind the case with the face, I’m thinking that a whole might be recognized as two even without explicitly or implicitly being aware of those possibilities.

## Is 1+1=2 Intuited?

Posted by Tanas Gjorgoski on December 2, 2006

Over at Brain Hammer, Pete Mandik as a part of his post Abusing the Being-Knowing Distinction proposed that a following principle can be valuable in attacking claims of intuited metaphysical truths:

It is intuited that P. Philosophical theory T explains P. But philosophical theory T is inconsistent with the fact that P is intuited. Too bad for T!

In the comments of the post I wondered if the following example would fall also under that principle:

1. It is intuited that 1+1=2
2. Russell and Whitehead gave theory of why 1+1=2 in Principia Mathematica
3. If 1+1=2 was true *because* of the logic in Principia, nobody could’ve possibly intuit its truth.

Pete raised the issue if 1+1=2 is intuited at all. I said I believe it is intuited, and Pete said that he believes that it is learned.
As I don’t want to abuse the comment-space over at Brain Hammer for discussing a question which is just marginally connected to the central topic of the post, I will put my thoughts on it here.

The question of what is a belief, and the issue of when we can say that one beliefs something is surely problematical.
When we say that a person P believes that X, we don’t e.g. expect that P continuously contemplates on the truth of X. Probably we will require only that in certain cases, in the right conditions, his belief that P to be actualized somehow, affecting what the person says, what the person does and so on. Based on that we can talk abut dispositional vs. occurrent belief. If that is so, then we can say that the person doesn’t have actually to claim that 1+1=2, in order to count as believing as 1+1=2. But if the person doesn’t claim that 1+1=2 (or sincerely report that he believes that 1+1=2), how do we get to know that person believes that 1+1=2?
One behavioral sign besides actually P claiming X, might be that P is surprised if he/she observes a situation in which X is shown false. While of course, this behavioral act doesn’t have to mean that P believes that X (generalizing beliefs to dispositions to act would fall into some kind of behaviorism), it is clear that it can be taken as a good indicator.

The question if this kind of behavior can be found in human infants has been tested by experiments. Wynn K.(1990)1 did experiments where he presented the following sequence to four and half month olds.

After this, two outcomes were presented, which are marked in the following picture as a)possible outcome and b)impossible outcome, and the time infants kept looking at the outcome was measured in each case.

In psychology, the sufficiently longer time in those cases is taken to represent a surprise. And that’s what Wynn got for the impossible outcome. The infants looked longer at it.

While I’m not very  knowledgeable about the field of developmental psychology, it seems to me that if those results are true and if the interpretation of the results is right, it isn’t very unreasonable to talk about infants having belief that 1+1=2. However because this is shown for very early pre-linguistic age, it makes it problematic to speak of a learned theory.

1Wynn, K. (1992) Addition and subtraction by human infants. Nature, 358: 749-750, reference taken from Lakoff&Nunez (2000) Where mathematics comes from p.16-17

Posted in Mathematics, Philosophy | 9 Comments »

## More on Hegel and Ratios

Posted by Tanas Gjorgoski on August 25, 2006

Some time ago, I had two posts (here and here) on infinite series, in which Hegel was also quoted saying that what is expressed in a ratio, can be only deficiently represented as aggregation (as infinite series).
While I tried to explain how that is so through an example, I didn’t try to give account of why it is so, which is of course more important that pointing to a fact. I will try to do this now, though in somewhat superficial way.

It has to do with the nature of the number…

Take for example some quantum… for example a distance in space. By itself that quantum is not determined as a number. It is not neither 1, nor 2, nor any other number by itself. It becomes number only in its relation to other quantum, to some other distance. So, it is their ratio which is 1:1, 2:1, 3:1, or any other. By, and within themselves both the distances aren’t any specific number. Only in their synthesis there is a number. Here we should be careful, and note that the fact that 1:1, 2:1 and 3:1, can be also presented by number as aggregate (i.e. 1=1, 2=1+1 or 3=1+1+1) in which the notion of ratio is left out, doesn’t mean that they are merely some kind of representations of that aggregate. On contrary, what is argued here, is that the number in its proper concept is always a form of ratio. So we can say that 1 is ratio of two qualities, 2 is also ratio of two other qualities and so on. Or said differently, the form of number as aggregate is abstraction from the richer concept of number as ratio. It shouldn’t come as surprise, then, that there will be a problem of expressing in a form of aggregate what is expressed in a form of ratio.

Further, concept of number as a ratio, also allows for natural connection to the rest of metaphysics in which it needs to relate to other “non-mathematical” concepts. And that is visible in the concept of measurement – there one quantum of some specific quality is compared with an another. The distance in space is quantum only as long it is compared with another distance, the amount of  time is quantum only if it is compared with other amount of time. Abstracting from the specific quality of the quanta compared (which is implicit in the act of measurement) we are left with the concept of number as ratio. What should be kept on mind in the measuring then, is that it is not a quantum as something in itself, but it is properly understood as quantum as ratio. Such understanding is implicit in all numbers which appear for example in natural science, and there too every measurement needs to be seen as a ratio (except in the counting of discrete things, where the unit presents itself as ontologically basic, i.e. – one thing, and where we can’t change it to something smaller or larger without changing the nature of the thing. For example, there is not much sense in counting in half-planets, or double-planets).

I should notice here that Hegel’s account of numbers isn’t left in this kind of “outside” connection to measure, which as I said at the start is somewhat superficial. In Science of Logic, the movement from quantity to measure, and further to essence is seen as a necessary resolution of contradictions that appear in those concepts when taken as separate and abstract. In the movement in Hegel’s Logic, number doesn’t stop at its proper understanding as ratio also, but is sublated further (brought into) in the concept of inverse ratio, and ratio of powers, before it “develops” into measure, and further into essence. But I don’t think I understand enough this particular development to write about it. If you are interested, you can check David Gray Carlson’s paper Hegel and the Becoming of Essence. (I had problems opening it directly from the web, but it opened fine after downloading it).

## Hegel and Infinite Series

Posted by Tanas Gjorgoski on June 28, 2006

Connected to the last post about comprehending how 1=0.99(9), here is what Hegel (§ 561 , The Science of Logic read at own risk) has to say about the issue of infinite series (in his own specific way):

Thus the usually so-called sum, the 2/7 or 1/(1 – a) is in fact a ratio; and this so-called finite expression is the truly infinite expression. The infinite series, on the other hand, is in truth a sum; its purpose is to represent in the form of a sum what is in itself a ratio, and the existing terms of the series are not terms of a ratio but of an aggregate.
Furthermore, the series is in fact the finite expression; for it is the incomplete aggregate and remains essentially deficient. According to what is really present in it, it is a specific quantum, but at the same time it is less than what it ought to be; and then, too, what it lacks is itself a specific quantum; this missing part is in fact that which is called infinite in the series, from the merely formal point of view that it is something lacking, a non-being; with respect to its content it is a finite quantum. Only what is actually present in the series, plus what is lacking, together constitute the amount of the fraction, the specific quantum which the series also ought to be but is not capable of being. The word infinite, even as used in infinite series, is commonly fancied to be something lofty and exalted; this is a kind of superstition, the superstition of the understanding; we have seen how, on the contrary, it indicates only a deficiency.

Let me try to explain previous quote with an example:

We say that 1+a+a2+…=1/(1-a) for a<1. But how to understand that addition of infinite number of terms? We can add as much terms as we like, but the sum will never be equal to the ratio. What does this equation say then?

If we subtract 1 from the  ratio (which is supposed to be sum of the infinite series) we have:
1/(1-a) – 1 = (1-1+a)/(1-a) = a·[1/(1-a)]  or by moving the 1 to the right side…
1/(1-a) = 1 + a·[1/(1-a)]

So, now we got to a recursive formula of the form
A = 1 + a·A  , where A is 1/(1-a)

And by iterating we get:
A = 1 + a·A
, then by changing second A with the whole right side we get…
A = 1 + a·(1 + a·A) = 1 + a + a2·A
, we change again, and we get…
A = 1 + a + a2·(1 + a·A) = 1 + a + a2 + a3·A
and so on, and so on ad infinitum.

We see that the sequence in the series can be generated from the ratio by recursive process, but it can never end, we can generate as much terms as we like, there will always be a certain left over (an·A), which also needs to be expressed through new terms.

So Hegel is saying that while we can represent what is specified by the ratio through infinite series, the series will be a “deficient” representation of what is clearly contained in the ratio itself.

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## Comprehending 1=0.99(9)

Posted by Tanas Gjorgoski on June 26, 2006

On first look it is weird when they tell you that 1 equals 0.999(9) where 9 is repeating.
Or, which is the same thing – that 3 * 0.333(3) , equals both 0.999(9) and 1.

I know I searched once on web for the explanation of why is it so – an explanation I could easily grasp/comprehend,  as intuitively it seems there is something wrong – our intuition tells us – “As much nines are repeating there, they won’t add-up to 1”. All I found is proofs that it is so, but from the philosophical point (well from common-sense point even) we can’t be happy (me at least) just with symbolic proofs, we need to comprehend the equality. So here goes the easy explanation…

When we have integer, let’s say four, we can say that we have a quantum of four things, a quantum which can be divided to four ones. Say you have four apples. You can divide them to four people – each person will get one apple.
Decimal numbers can be comprehended in similar way, just that if we add decimal fractions, we need to imagine that even the ones are further dividable – that they are made from parts. We need to imagine each one as made from ten parts, and each of those ten parts as made from even smaller ten parts, and so on…

In such way 1 would mean the item, 0.1 would mean one of the ten parts of the item (1), 0.01 would mean one of the ten part of the 0.1 part, and so on…

Now, let’s try to divide one item to three people. Because the item consist of ten parts, we will give three parts to each person, and we will be left with one of those ten parts that we need to divide:

1/3= 0.3 + 0.1/3

This 0.1 is again consisting of ten parts and we will likewise divide it between those three people – three parts to each, but we will be left again with one (now ten times smaller) part to divide further…

1/3= 0.3 + 0.03 + 0.01/3 , and dividing further…
1/3 = 0.3 + 0.03 + 0.003 + 0.001/3 , and so on..

Or if we sum the parts which are already divided we get

1/3 = 0.3 + 0.1/3
1/3 = 0.33 + 0.01/3
1/3 = 0.333 + 0.001/3 etc…

We see that this adding of cipher 3 to infinity, is not consequence of the part which is already divided, but of the part (left-over) which is left to be divided. As many threes are generated there will be still a part to be divided which would generate new threes. We can now say the following… the infinity in the never-ending repeating cipher 3, is not about the infinity of how much ciphers 3 are generated, but about the impossibility to finish the dividing process.

So, it is the wrong comprehension of what 0.333(3) means, that is causing the intuition against 1 being equal to 0.333(3) * 3. If we imagine 0.33(3) vaguely as some never ending string of cipher 3 – a decimal number in decimal notation that just has infinity of ciphers, our intuition tells you that however far you multiply with 3, you will only get 0.999(9), and however far you go, it won’t be 1.
But comprehending 0.33(3) in that way, is ignoring the reason of that infinity, for example in the given process – ignoring the left over which is still to be divided that causes us to add new and new ciphers 3, and which will never end. So, we need to see repeating cipher as a new notation added to the decimal system, which tell us about that impossibility to finish the description in the normal decimal notation, and not a shortcut for normal decimal notation, in which we could write the cipher 3 infinite number of times.

By the way we can divide 1 in similar way to get 0.99(9) directly…
We divide 10 of the parts (0.1) to 10 people, so that we divide 9 or those parts to 10 people, and leave one more to divide; and that we repeat that:
1/10=0.9/10+0.1/10=0.9/10 + 0.09/10 + 0.01/10, etc…
We will get 1 = 0.99(9), again this repeating 9 signifying the impossibility to finish the process.

* It isn’t really necessary for each part consist of 10 parts. What we need is just that each part can be divided to 10 parts, and if it can, we can treat it as consisting of 10 parts. For example if we don’t have quantum, but a continuous magnitude, it can be divided in any number of parts we want, hence to 10 also.

Posted in Mathematics, Philosophy | 21 Comments »