Tegmark's Mathematical Universe

Surely we must mentally distinguish the underlying reality from our symbolic representations of it? Mathematical symbols, statements and graphs can REPRESENT the underlying reality, EXCEPT for the precise "random" details of quantum processes, which seemingly cannot be mathematically represented. The underlying reality seemingly does not conform to scientist's expectations of a completely mathematical universe.

But what underlying reality do physicist's mathematical statements (and more complex derivatives and graphs) represent? Should we say that e.g. "a bc = d" is the type of thing that represents the fundamental underlying reality, OR should we say that the above represents 3 aspects:

1) a b c d

2) - * /

3) =

of the fundamental underlying reality?

Lorraine,

Hi. I totally agree with you. IMHO, it's very important to distinguish the mind's conception of something from the thing itself. As you said, mathematical symbols, which our in our heads, describe an underlying physical reality but are not themselves that underlying reality. And, just claiming that mathematical constructs exist in some Platonic realm doesn't explain anything. Please show me this realm now. Point it out. Until then, this argument that mathematical constructs exist outside our heads and are themselves physical reality is nothing more than the God argument. It may be correct but can never be tested.

In regard to a +bc = d, I'd say that your three choices are correct. To me, one has to start with an existent state. That is "one" existent state. Add another existent state, and a mind could say: one existent state plus another existent state gives two existent states. That is, math is just a mind describing, after the fact, physically existent states.

Peter Jackson, above, suggests that we can't even accept "a = a". I agree with that, too, in that I think there can be different and opposing ways of perceiving the same existent state, a. Looked at the same way, a would equal a. But, looked at in a second way, a might not look like a as perceived in a first way. I use this type of thinking in my argument in an earlier fqxi essay contest that "something" and "nothing" are really two different ways of describing the same underlying physical reality, the absolute lack-of-all. If anyone's interested, more of this is at my website at

sites. google.com/site/ralphthewebsite

Thanks!

Lorraine,

"Surely we must mentally distinguish the underlying reality from our symbolic representations of it" Lorraine, you've hit the nail on the head. I just posted something on topic 1229 -Killing Time- before realizing the thread had been dead for 7 months. I re-post it here because it discusses this underlying reality you speak of.

..." The problem is always the same. Not knowing the difference between physics and metaphysics, and the limits of each.

There is EXPERIENCE, what we make up of the world through our senses and mind. EXPERIENCE requires our presence to happen and to exist. So, forget about physics without observer; we are always part and cause of the observation and its interpretation.

And there is SUBSTANCE, that which does not require our presence or observation to exist.

The real universe is made of substance. Our reality is made of our experience of this real universe. We create our own reality which, without our senses and mind, is just a dark place filled with matter and radiations. (Even " matter and radiations" are evolved concepts of ours).

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

The thing is that we can only know about time indirectly by deducing it from experience like motion, change, etc. The reason that time is so elusive is that it is not physical but rather metaphysical ; it cannot be an experience. It is here and there without requiring our presence. We may only deduce or infer the passage of time from the experience of change.

----- The passage of time is metaphysical and it has to be treated as a substance.(opposite of experience)

When we confront the existence of a substance with the requirements of operational logic, we find that there can only be one substance making the whole universe. Science has effected great stride in the required reductionnism; mass and energy, time and space, electricity and magnetism... But we are facing a wall because time is involved and appears necessary in all of them. If they exist, they are just various forms of the passage of time.

The important things to remember are 1) The passage of time is metaphysical (a substance) 2) It is a process that is dynamical in nature 3) it is a spontaneous process driving all other spontaneous processes. In this, it is the simplest and most basic process making the universe.

For more details, see my previous two FQXI essays."

---- You see in the text above that the basic workings of the universe are in very simple operations of logic (which are at the root of mathematics). Logic, in its prime, is essentially the rules that govern our expectations we have about outcomes that we have acquired from experience in our reality. Logic is scale independant (works for atoms or galaxies) and is system independant (the basis of all systems is logical consistency).

---- As for maths, I have discussed earlier, "natural addition", or adding in a geometrical way as in "moving objects closer together to really add there intrinsic properties i.e. mass etc." . In school we used to learn addition by grouping and piling together objects. So does a planets gaining weight, by aggregation, not by calculations.. It is easy to recognize the difference between this natural addition, which the universe performs all the time, and adding money in my pocket with money in my bank account.

Marcel,

Hi Roger and Marcel,

Thanks for responding to my post.

I guess what I was trying to say (very badly) is that seemingly much of the underlying reality can be represented and its behaviour predicted via the various mathematical formulations of Laws of Science/Physical Law/Laws of Nature.

You can argue about which mathematical statements are the ones that correctly represent reality, but the point is that our representations of the underlying reality seem to take the form of mathematical equations. Something about mathematical equations represents the nature of reality!

If there is no Platonic Realm, and I think there can be no such thing, then everything exists within this universe. So I was trying to ask: is a whole mathematical equation the type of thing that represents the underlying fundamental reality? Or do the common component parts of mathematical equations, like + - / * and the equal sign, represent underlying more fundamental realities? Obviously these necessary symbols would represent non-measurable realities whose existence can only be inferred.

An advantage of the component parts view seems to be that numbers founds in nature, instead of being seen as mysterious Platonic objects, might be seen as being constructed out of the same component parts as Laws of Nature. E.g. if we say that "a + bc = d" represents a Law of Nature constructed out of component part fundamental realities, then "(d +d + d)/d" represents a number similarly constructed. The component part view implies that that numbers found in nature have a hidden structure, but a structure that may be impossible to determine: there are many possible ways to construct the number 3 using mathematical symbols.

Lorraine

Lorraine,

"Something about mathematical equations represents the nature of reality!"

Marcel: It represents the behavior of nature as seen from our reality.

": is a whole mathematical equation the type of thing that represents the underlying fundamental reality?"

Marcel: Yes, in a way. It describes accurately how we perceive the behavior of the universe. But the equation does not run the universe nor does it hint at what does.

"Or do the common component parts of mathematical equations, like - / * and the equal sign, represent underlying more fundamental realities?

Marcel:

I think I may partly address your question in the following....

... Consider E=MC2 . At first glance it is not logical because we multiply or equate elements that are or of different nature; mass, speed , energy.. We get away with it because it works for its purpose in our reality which means that these different elements DO have logical equivalence in the underlying reality. But operational logic cannot accommodate more than one (type) of substance of one nature. (The proverbial apples and oranges). In my theory (see essays) this unique substance is the passage of time (a process). This means that all the elements of the equation represent each a concept, form or variation of time. This way the "equation" could become logical. The problem is figuring out which concept, form or variation of time they represent and how or whether the - / * are or represent valid logical operations.

The equal (=) sign is for accounting a static equivalence. In the underlying reality, operations are spontaneous and logically driven and are transformations as in "before" and "after". So, an equation (=), even "logical" would remain a sentient accounting, not a natural process that runs the universe.

Marcel,

Lorraine,

Hi. I'd agree that everything exists within the universe. My view is that for the universe to be here at all, there must have been some indivisible (e.g., not further subdividable) physically existent state. Whether this state is called a particle, mathematical construct, law of physics, etc., it doesn't matter. It's some existent state. This state must have a way of creating additional states that can interact physically to produce the motion, forces, etc. that are in our universe. Then, after the fact, humans came and invented mathematical symbols and rules to describe these existent states and their interactions. So, I'd say that the type of math that represents the most basic underlying reality is actually the number 1. "One" represents the presence of a single existent state. Later, humans than invented addition, substraction, etc. to represent the presence of additional existent states, removal of states, etc. My vote is that equations are way down the road from this. I'm not sure if this is related to your component parts view?

Anyways, that's my vote. Thanks!

Marcel and Roger,

When I wrote "underlying fundamental reality", I didn't mean to imply some sort of ultimate fundamental reality or some original indivisible state. I just meant the fundamental information and relationships that science has detected, and has represented symbolically as mathematical equations (Laws of Nature).

Does Marcel's example of a Law of Nature represent an indivisible reality, or does it represent a whole that is somehow constructed from particular aspects of reality: energy information represented by "E"; mass information represented by "m"; (perhaps) relationship represented by the implied multiplication symbol "ﾃ--"; (perhaps) balance or transformation represented by "=" etc. If "E" and "m" represent fundamental aspects of reality, why wouldn't the necessary symbols "ﾃ--" and "=" also represent fundamental aspects of reality?

Why is science seemingly blind to these symbols? Is science avoiding facing up to essential, but non-measurable, aspects of reality?

Lorraine

Lorraine Ford (feb 21 22:59),

Your question about the relation between primary reality and algebraic equations is really amazing. Indeed, I realize we practise mathematical equations only to compare phenomena and their relations (empiric science). Do we need equations when we describe the primary reality? No, you are right. Describing something "absolute" (an infinite whole) don't involve equations.

Your next question is: "Why is science seemingly blind to these symbols? Is science avoiding facing up to essential, but non-measurable, aspects of reality?"

No, I don't think so. I suppose everyone likes a reality that is "comfortable". A reality that looks like an altering image. You can imagine it, you can discuss it, you can draw it. Personally, I don't think a mathematical universe is a pure algebraic universe. We use algebra to calculate. So an algebraic universe have to represent a geometric universe: just like our daily world. There is only one problem: we cannot observe this reality with our senses/instruments. So the mathematical reality forms general relativity and quantum fields too.

For scientist, the most important question about the mathematical reality is: "How can I explore this reality? Where do I begin?" The answer is obvious: foundational mathematics. That is logic and set theory. So don't make the unrecoverable mistake to start describing a detail/phenomenon somewhere in the universe (like string theory). You have to describe absolute space and absolute time. Anyway, it isn't difficult. You have to choose the right starting point.

Henk

2012-02-24

Lorraine, Roger, Henk,

How do I start? Step back and look at the whole picture. What are we missing? Oh! Yes! Nowhere do we admit that anything exists by itself. Physics is about our relationship with this universe. What do I use then to explore or address the non physical? Mathematics? Hum.... It is numbers... involves quantities and measurement ... then the observer is still there, involved.. No! Something more primitive than mathematics yet eminently connected to it; logic! To look at quantities means measurements and the intention to describe how things work. That is nice, but the universe has no need for measurements. The HOW question is for our intellect. The universe evolves spontaneously so the real question is WHY?

So, we have the question 'why" and a tool that is simple logic. This would call for metaphysics but this field is in a poor state and is still a system of opinion. In order to go somewhere with this you have to work with a truth system, a system based and made of inferences drawn from a fundamental impossibility. Euh... the rule of non-contradiction is definitely the right one, as the starting rule of impossibility! As case in point all other truth systems abide by it, maths the first one.

Beyond and as basis of our perceptions there must be something that really exists we will call a substance. This substance could undergo logical operations but only on itself or variations (still of the same nature) of itself. For these reasons and a few cosmological pointers we will accept that the whole universe is and has to be made of this single substance. We are in business!!

Now, logic is taken here as having its full meaning. First it describes a valid operation on the substance like addition, substitution etc. .... Secondly, logic describes a spontaneous becoming i.e. the logical spontaneous outcome of a given state or situation. This spontaneous logical operation is the CAUSE. We have now recovered both the SUBSTANCE and CAUSE of the foundation of metaphysics. After some work, you should find out that there can only be one single type of logical cause in the universe. And this cause HAS to be built into a property of the substance; it can`t be otherwise. And this substance and this cause are.....

Marcel,

Henk,

You refer to mathematics, algebra, logic and set theory as though you think mathematics can be taken for granted: something that is assumed to exist that requires no further explanation. But surely "the right starting point" is to ask what mathematics is, and where it fits into the scheme of things? I think 4 points about mathematics are relevant:

First there is the question of a Platonic realm. I start from the assumption that there is no Platonic realm - that is, I think everything exists within this universe including mathematics: mathematics is part of the universe - it is not external to the universe. Unfortunately, even physicists who profess to not believe in a Platonic realm seem to treat their mathematical equations (laws of nature) as though they were external to the universe i.e. things unable to be explained, or things not requiring explanation in the context of this universe.

Second, I assume that mathematics arises out of the reality of living things: mathematics started out as, and continues to be, a system of symbolic representation. We are made out of the underlying reality of the universe; we are not separate from this underlying reality: I consider that mathematics could only make sense to us if and only if the basic elements /symbols of mathematics faithfully represented corresponding basic elements and relationships in the underlying reality. So there was no mathematics in the early universe - there was only the underlying reality, much of which we can represent using symbols*. These symbols include "+ - ﾃｷ ﾃ--" and "=", which presumably represent aspects of the underlying reality in this universe, not things that exist in a Platonic realm. These symbols, alone or as part of "law of nature" mathematical equations, seemingly represent aspects of reality that are not measurable, whose existence can only be inferred.

Third, while the basic elements/symbols of maths can be utilized to represent the underlying reality (e.g. as laws of nature), mathematicians or physicists can also creatively assemble mathematical symbols to represent artificial realities: this is similar to the way words in a story, or colour and line in a painting, can be used to represent actual experienced reality or to represent imagined, artificial realities**.

Fourth, the above views might lend support to a view that numbers found in nature, instead of being seen as mysterious Platonic objects, might be seen as being constructed out of the same "component parts" as laws of nature. E.g. if we say that "a + bc = d" represents a law of nature, then "(d +d + d)/d" represents a number similarly constructed. This view implies that that numbers found in nature have a hidden structure, but a structure that may be impossible to determine: there are many possible ways to construct the number 3 using mathematical symbols.

Lorraine

* But seemingly not all aspects of the underlying reality are able to be represented with mathematical symbols e.g. the precise "random" outcomes of quantum events are seemingly not representable as one side of a mathematical equation.

**This view would indicate that mathematics is both discovered AND invented: the type of symbols used in mathematical equations, and the form of a mathematical equation, represent discoveries about aspects of the nature of reality; but mathematics is invented in the sense that it is a creative reworking using these types of symbols.

Hi Marcel and Lorraine,

I agree, physics is in part about our relations, as observers, with the observed universe. It is also in part about the relations of the material universe with itself as it is those relations that provide the forces for continual change (including the production of information that we are able to receive and interpret).

I like what was said about non contradiction. However contradiction can also just be the product of looking at things in a particular way, an opinion. Like the impossible triangle optical illusion. Or a flower that is patterned and not patterned, depending upon whether seen by a human or a bee. It is so with the incompatibility of QM and Einstein's relativity IMHO. Contradiction may be indicating a wrong idea and -or- a wrong or limited perspective of the observer; (To do with lack of complete information). Both are important in illusionists' work.

Lorraine,

IMHO the key to understanding the universe is a*b, the operation of multiplication between two algebraic elements. One cannot divorce algebra from mathematics, and it is *algebra* which defines the operation of multiplication between algebraic elements, one of three required operations with element addition and element multiplication by a scalar.

The operation of multiplication is not singularly defined, since there are many different algebras, each with their own definition. So there is a choice to be made on which algebra most succinctly matches up to physical reality. The other two required operations for algebra are commonly defined for all algebras, which is why * is so important.

In my essay for the last FQXi contest, The Algebra of Everything, I present a brief picture about why physicists need to take more notice of Octonion Algebra. The variability of how one may define * within Octonion Algebra and the realization it must impose no variability on the equations of physical reality *demands* Electrodynamics is the way it is, and provides sufficient dimensionality to include a potential theory of Gravitation in a natural coexistent way when the proper generally covariant definition for differentiation (provided) is used. Take a look if you have not already.

Rick

Dear Marcel,

you said .....

"Now, logic is taken here as having its full meaning. First it describes a valid operation on the substance like addition, substitution etc. .... Secondly, logic describes a spontaneous becoming i.e. the logical spontaneous outcome of a given state or situation. This spontaneous logical operation is the CAUSE. We have now recovered both the SUBSTANCE and CAUSE of the foundation of metaphysics. After some work, you should find out that there can only be one single type of logical cause in the universe. And this cause HAS to be built into a property of the substance; it can`t be otherwise. And this substance and this cause are....."

You aimed exactly right.

This substance and this cause are random numbers and imperative (only possible ones) logical operations on them. Yep, That is all to my theory which recreates all the physics we know.

http://www.qsa.netne.net

more detailed explanation are in posts 20,25,43 in this thread.

http://cosmoquest.org/forum/showthread.php?140913-%93Reality-is-nothing-but-a-mathematical-structure-literally%94

Lorraine (feb 28, 00:43),

Sorry, because of the misunderstanding; I was a bit vague. And to rectify: I don't take mathematics for granted. In fact, throw away all your books about mathematics because they are largely useless if you want to explore the mathematical universe. Most mathematics is some kind of applied mathematics. You need foundational mathematics or meta-mathematics.

[First] The Platonic realm was a fascinating new hypothesis about the nature of reality some 2500 years ago. Like string theories... (nice but at the long last useless because there are too much assumptions)

[Second] Nature forms atoms, quantum fields, chemistry elements and... humans too. We - humans - do not form nature. So there was math, there is math and there will always be math. With or without humans. (The laws of) Mathematics is the structure of the universe. But the "fabric" of the universe is unknown and will stay unknown. It is impossible to detect or deduce what is everywhere the same (we can only detect or deduce differences).

[Third] Look outside your window. Probably you see a house at the other side of the street. That is composite geometry (macro cosmos). And in the micro cosmos everything is geometry too (but on a fundamental non-composite level). So, no symbols, no mathematical notation system ( a sigh of relief ;-).

[Fourth] Numbers are representing objects (phenomena) and not vice versa. So there are no existential numbers to be found within the mathematical universe. Within the mathematical universe numbers represent mathematical objects and mathematical proportions. Just like we use to do in physics.

And the right starting point: you have to analyse the scope of reality (what is real and what isn't real). See the reply of Georgina Perry (feb 28, 09:03): you have to choose the right perspective as observer and explorer.

Henk

Hi all,

First, I think you are making a big deal out of nothing. People believe in that mathematical facts are true, the only controversy is why. Even that, most philosophers of mathematics will tell you that there is no need for the why, it is obviously just is.

Second, we deduce facts that we believe in, even though sometimes we don't actually touch or see them. We fully believe in the electron even though nobody has actually seen one. Also, we have not seen an actual circle either; its curved line width goes to zero. So fundamentally there is no barrier to deduce the ontology of reality.

Third, we use symbols to represent things like a circle. But again if no humans existed a circle will be there.

Forth, we do use objects to represent mathematical objects. Do you remember when you counted numbers with your fingers when you were a little kid? That's how we conceptualize mathematical facts with our brain.

Sure, clever conjectures by Tegmark and others have considered the possibility, but there has not been any credible theory that really ties reality to a mathematical realm, not until my theory that is.

Mathematics created our reality, and we are the living proof in the platonic existence of reality.

See my reply to Marcel.

Henk,

In [Second] you say "(The laws of) Mathematics is the structure of the universe". But the "random" outcomes from quantum events seemingly indicate that the reality that can be represented by mathematics is only part of the story. These random outcomes cannot precisely be predicted i.e. they cannot be represented as a (law of nature) mathematical equation. Mathematics is seemingly useless in predicting the precise individual outcomes of quantum events, so I don't think you can make an unqualified claim that "(The laws of) Mathematics is the structure of the universe".

You conclude: "you have to analyse the scope of reality (what is real and what isn't real) ". So, looking at the reality that CAN be represented by law of nature mathematical equations, we can assume that, thanks to the work of physicists, law of nature mathematical equations represent what is real, what actually exists within the universe.

What I'm trying to say is not necessarily about the precise details of individual law of nature mathematical equations: I'm saying that the form of a mathematical equation represents something about the nature of reality. The symbols "+ - ﾃｷ ﾃ--" and "=" are necessary parts of law of nature equations. My main point is that, just like the symbols "E" and "m" (representing information about energy and mass respectively) represent acknowledged aspects of reality, the symbols "+ - ﾃｷ ﾃ--" and "=" in these equations represent evidence of unacknowledged aspects of reality.

The symbols "+ - ﾃｷ ﾃ--" and "=" seemingly represent aspects of reality that are not measurable, whose existence can only be inferred. Science does not acknowledge that there are aspects of reality that are not measurable. If there is no Platonic realm, then these symbols represent fundamental aspects of reality that are part of this universe. Surely physicists cannot form a correct picture of reality unless they acknowledge, and attempt to analyse ALL aspects of fundamental reality, not just those like energy and mass that can be both measured and represented.

Lorraine

Lorraine (march 2, 11:37),

I understand and agree with your explanation about the potential relationship between mathematical operations (algebra) and the laws of physics. But a big part of the problem is the impossibility to understand the mathematical reality by using the descriptions (equations) of phenomena in physics. And that's because physical phenomena do not represent the primary level of reality. So you cannot build yourself an image of this underlying reality (and that's the problem in quantum field theory too).

Your example: E = mc2.

This equation means: the invariant mass of a system is the total energy divided by the speed of light squared. Now the meaning of this equation "within the mathematical reality".

E is the amount of moved area into the volume.

And... everything is clear? Of course not. This is another kind of incomprehensible "speech".

So trying to explore the mathematical reality has nothing to do with translating applied mathematics and theoretical physics into this mathematical model of reality. It is just the other way: exploring the mathematical reality will solve all those incomprehensible theories that are composed by physicists and mathematicians.

We have to accept the fact that empiric science is founded by the process of analysing (composed) phenomena. These phenomena are formed by one underlying structure of reality (that's the conviction of most scientist). So, we cannot explore reality by comparing physics and mathematics. That's not the way. You have to "build" reality from the bottom up with the use of mathematics. And of course: that's (some kind of) logic and set theory (the proposed foundations of mathematics).

Physicists have tried to form a correct picture of reality by attempting to analyse all aspects of fundamental reality, but they didn't succeed (e.g. string theory and super symmetry).

Henk

Lorraine,

..Relationship with the universe... Everything we know we learned from our interaction with the universe in the capacity of an observer. If you think or look at two particles about to interact, you are an observer. In that capacity you may think you are not involved in what is observed. This is wrong in many ways. You are a third entity but you are not part of the equation; what are you?. And yet, what you look at is highly transformed by using sense and brain. The point of view of the observer, a third party, is in no way a part of the universe. How could the universe be made of a point of view if we only showed up recently in this universe? To understand the universe you have to leave your special position of observer and take the place of one of the two particles. Now you may be part of the action in the capacity of an actor. ... In such a place there is no numbers because you don't "know" what is around you or away... You only feel whatever where you are ... no numbers... no equation .... You exists, that's all. And existence is the most basic property to consider.

Georgina,

Non-contradiction (NC) is the logical basis of everything, maths ,science . .. Illusion is perception, as is our reality. GR and QM are both right as different points of view (truth systems) and as such can not be reconciled. Consider that the point of view of each actually look at the same thing. Then, only in the actual subject of their observation can there be unity i.e. the underlying reality.

Qsa,

Although a unique substance makes for an incredibly efficient use of numbers they must support an actual substance that exists by itself. The universe is not made of numbers. There must be a logical primer in order to get everything moving. Numbers are for people who wants to know "how much". The universe only need a "why" to get moving. (I will check your site -thanks)

Lorraine Ford wrote on Feb. 28, 2013 @ 00:43 GMT

You explain in more details your case/theory. Yes, maths being born out of logic do relate to actual logical operations the universe performs in the underlying reality. But all the metric aspect is for your own intellect - need to know. The universe only needs a logical reason to evolve. We may sum up and quantify these logical reasons in one place in order to come up with a "law of nature" that describes or match our "observations". But Nature does not follow the law; it follows the logic.

Rick,

The question could be: Does nature effect a *? How does it do that with a substance? What is the greater naturalness of octonions? Better description or better understanding?

Henk,

"But the "fabric" of the universe is unknown and will stay unknown." Well, we have learned over the years to regard "never" statements with a certain smile... We can deduce what the substance is. The (scarry) part is: do we really want to? Are we ready and mature enough to do so? Questions for which we really don't want the answer are typically not being asked.

qsa qsa replied on Mar. 1, 2013 @ 00:24 GMT

qsa,

"But again if no humans existed a circle will be there." Nope. WE make the circle full. By perception we integrate and consider a bunch of points and make the circle. AS an observer you take the picture and make it as a circle. As an actor you would be one of those points not knowing anything beyond just you.

Physics tells you the same thing. Because the speed of light has a limit, these points can't interact(1) with each other simultaneously i.e each point is in fact away in time from each other. Therefore, no two points of this "circle" are at the same moment! Your privileged position as an observer may allow you to see them all at once in a (same) moment of perception. But they are not at the same moment to form a "circle". They are a sequence of points accross time all having A same time distance to each points of (central) line. (Otherwise, they could form a line drawn on the surface of a sphere) .

(1) To involve the "interaction" between points detaches you from a spectators point of view and gives you an operational point of view.

Marcel,

Marcel,

thank you for your reply. Yes I agree, there is no contradiction in the underlying reality because it is everything -that it can potentially be observed to be- simultaneously, every viewpoint. The impossible triangle has the potential to be seen as a triangle and not be seen as a triangle without contradiction . The flower has the potential to be patterned and not patterned simultaneously. The magicians hat has the potential to be regarded as empty or not empty (depending upon whether seen with visible light or as an X-ray image). This is important in relation to wave function collapse IMHO as the potential for many different observed manifestations becomes one particular manifestation upon observation.