How much can we push for consistently answering deep questions? At least based on my efforts!

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Above wallpaper reference¹

While I was writing the Geometrical probability² I notice I need to explain the main reason why I prefer Geometrical probability more than traditional probability theory. Additional to the better understanding of probability, Geometrical Probability has a very awesome feature, but I have to explain and complain about the mathematics', and physics', foundation a little bit!

By the way, for completeness let's mention that I consider my posts as full-fledged scientific articles,

If you are like me, educated in a university, they have taught you we build mathematics on top of the Set Theory³, and especially its most common axiomatic set theory ZFC. You have probably heard of Russell's paradox ⁴, which is kinda resolved, but it's not satisfying! I have multiple arguments about their "choice" though, that I am going to share here.

Lets starting by the definition of the set. A set is a collection of objects, which we call them its elements. Usually we don't need to mention that a set doesn't have "order", since "order" is an additional ingredient that we refuse to add to the definition of the set. But is it? It's more clear when we add "order" to define a new object by labelling elements by indices. We call it an array or a list. So by definition set has fewer constraints, therefore, it's more qualified to be primitive, which means it makes more sense to build mathematics on top of sets instead of arrays, right? Wrong!

First, remember it's all about relativity! In the human mind, it looks like naturally we need to do something to add "order", so humanity developed languages that "order" is an additional thing that you can add. For instance, you can add order to your room by tidying it up. You basically need energy to combat the Second Law of Thromodynamics. However, from the nature's perspective, everything has "order", so it is continually removing "order", again recall Second Law of Thermodynamics! Thus, we have naturally fell into yet another language trap by thinking "order" is a constraint that should be added later! But in reality the consistent approach is that the "disorder" is a constraint that should be added later into the ingredients. The language is so not built for it, that I doubt "disorder" is the right word for "not being ordered", but I will go with that anyway!

You can also see it in the computer science. In computers, creating an array is cheap, and trivial, but to make a set, that doesn't duplicate since they don't have "order", you need to map the elements into some hash tables, or a tree graph. Guess what! Both hash table, and the way we create the tree graph, rely on arrays. Therefore, from the computer's perspective, or you can say in reality, "order" is cheaper to achieve than "disorder"! The "order" comes first. The array is more primitive than the set.

Additionally, just look at the simplicity of Peano axioms⁵. They are much more trivial than anything ZFC suggests. You build natural numbers, by using a recursive function⁶, and define array with natural numbers as indices. Done! This is the foundation that makes sense to me. You can rewrite all the useful theorems in mathematics by using arrays instead of sets, just be aware you need to handle duplications since sets are doing that magically under the hood. This is so good, since you cannot build array of Real numbers, and we have proved that Real numbers are paradoxical ⁷, so a foundation on top of arrays will stop us to make such centuries of wasting times on Real numbers. After all Real numbers are not real!

These days, the Youtube is filled with videos about Axiom of choice⁸, one of ZFC's axioms, and the surrounding discussions. So it's perfect place and time to debunk it! People usually have problem with the axiom of choice when it's going to choose in a set with infinite elements. However, can you choose an element out of a non-empty sack of finite elements that doesn't have order? When you put your hand inside the sack your hand has a position, thus, when you pick the element, you have chosen it, it must be in an order respect to other elements. It's like saying in computer science, you can implement the "popFirstElement" method for both of HashSet, TreeSet, or any other implementations of sets since they are all need to be implemented on top of arrays. After implementing the "popFirstElement" you proved that there is an order, so what you have is not a true set, that doesn't have any "order". This all implies that you cannot choose an element even from a finite non-empty set, so no need to try to choose from the edge case of a set with infinite elements. The "lack of order" and "choosing" are in contradictions. Therefore, the "set" and "choosing" are in contradictions. Thus, basically we debunked all the theorem of mathematics that have been using sets, since they have to choose something out of their sets! Not too fast, since we can rewrite most of them with arrays, with little trade-off.

You have probably noticed that usually I am asking a lot of questions, and I demand consistent answers, and I stick my mind to the answers. By the way, based on answers that I found, I can claim a lot of times human's brain likes the game more than the answers! As it's known, but I cannot find the reference, gamblers are not seeking the rewards, even though they pretend in that way! Most of all, they don't want their game to be finished! After all, the brains of all human beings are very similar!

So next question for me was: why all the clever mathematician in the history didn't have answers you have? There must be some principles that they followed, where lead them to this position, right? It's coming back to the idea that people will fit into the system, so we shouldn't try to fix people, but fix the systems. In pop-culture words, we should design the system in a way that even bad people find right things profitable! For instance, a mathematical system on top of arrays, instead of sets, would have stopped us waste a lot of time to build 1+1=2 as Bertrand Russell⁹ did in his wonderful work, Principia Mathematica¹⁰, even though, without his awesome work I couldn't come to my current conclusions, but still! He's a true hero.

Sometimes I define mathematics as symbolic philosophy, to show that some mathematicians are seeking freedom, so they don't mind if an object, that we call set, has magical properties, such as "not being ordered" and "being able to choose its elements" together. I call it magical since they avoid implementing how it could work. Some of them are so free that they are called Platonist. They are so free that they don't have to pay for the computation costs! The final answer exists in another universe even before you start computing! We just need to "discover" the answers!

Anyway! At some point you are getting sick of that freedom, since it brings all kinds of contradictions and paradoxes! That kind of freedom is beyond me! You want to avoid contradictions and paradoxes, since they are painful when you apply them to our real universe. Therefore, most of the time I define mathematics as a symbolic construction, since construction is proof, which will exactly avoid any contradictions and paradoxes as I desire. I am kinda referring to the Curry–Howard correspondence¹¹. The latter definition is my favorite one of course. Happily, I am not alone, and we have Constructivism¹², even though there are differences! For instance, as far as I know, replacing sets with arrays my have suggested, but the next principle below is totally mine. Thus, I have a variation of constructivism here, which I call it Constructivity, since it's not only mathematics and on its physics tail it's following Einstein's Relativity ¹³. It's the notion that universe is following both Relativity, and Constructivity. So the first principle of Constructivity would be: construct everything on top of Natural numbers, arrays, and recursive functions⁶. Notice, the recursive functions are secretly the same as Morphisms¹⁴ in Category theory ¹⁵. The Constructivity has more principles, so I have to explain a little bit more.

Since you cannot construct everything on top of only numbers, arrays and recursive functions! You need more axioms, thinking about Gödel's incompleteness theorems¹⁶, then what's the guideline to find the best array of axioms? The old answer to optimizing the axioms is the Occam's razor ¹⁷. It suggests the system that is constructed with the smallest possible set of axioms should be the way to go. Which in the current time its loophole has been exploited, when people started to invent magical axioms out of nowhere to build the minimum set of axioms required by this principle. Until this point we already explored sets and Real numbers, as the examples of magical axioms. However, it leads us to disasters like the Copenhagen interpretation¹⁸, and even String Theory!

Just look at the cost we are paying to quantum computers based on false promises of Copenhagen interpretation¹⁸! Of course, collapse of wavefunction is not instantaneous! It's clearly following Heisenberg's uncertainty principle¹⁹ since \(\Delta \nu\), uncertainty in frequency, increases by increasing the temperature, recall Planck's law ²⁰, thus \(\Delta t\), uncertainty in time, will decrease. Thus, it's not instantaneous, therefore, quantum algorithms will take much much much more time to run than what's claimed by the Copenhagen interpretation. By the way, this is a known fact that the wavefunctions will collapse faster if we increase the temperature, but the quantum computers's hype is flourishing! Imagine the amount of the hype if I can debunk a billion-dollar industry in a paragraph, in the corner of my thought! This is the pain I am talking about when I say "certified" bureaucratic scientists ignore the contradiction and paradoxes!

Thus, the second principle of Constructivity is: additional to optimizing axioms to be minimum, the axioms MUST be observable. I will call it Observability principle. For instance, all three principles of Special and General Relativity ^{21 22 13} are satisfying this Observability principle. As mentioned, Constructivity is trying to follow Relativity. We want to gain the Einstein's power to predict what MUST be real! I don't want to repeat the stories here, but it happened! Einstein didn't invent this power! It was there. He just became the master of this power. It was not just Einstein, Faraday had this power, even though nobody considers him as a mathematician! I also love Clausius' logic based on the observations to come up with his theorem ²³. Basically all of them, Newtonian mechanics, Theormodynamics, and Electrodynamics, have been following Observable principles. Until Copenhagen interpretations of Quantum mechanics ²⁴ broke the rule by claiming that if you observe you will not see the wavefunction, what's defined on its other principles! Don't let me start on the strings in the String Theory, also loops in the Loop quantum gravity, etc.

We can conclude that Constructivity has the following principles by polishing what we had above:

• Construct everything on top of Natural numbers, arrays, and recursive functions ^{6 14}.
• Construction is a step-by-step evolution by using recursive functions.
• The reality, as well as the observations, aka the measurements, are all constructable.
• Additional to optimizing axioms to be minimum, the axioms MUST be observable.

Thus, the Geometrical probability ² is an attempt to use geometry instead of sets to describe and compute probabilities.

In mathematics itself, set is not observable, and Real numbers is not observable, which are the source of most of the paradoxes in our current mathematics.

Additionally, since we used Differential geometry ²⁵ in the Geometrical probability ², it's important to mention it's all constructable, until you try to construct the global picture of Fiber bundle²⁶! So non-constructable inventions are here and there in the realm of mathematics.

Even though we can technically put all the Rational numbers into an array by diagonally picking them, but they will not be sorted! To have a sorted one, we use a sub-combination of the entire Rational numbers by only considering the ones that spaced equally. Then we can call that spacing the accuracy, \(a=b^{-d}\), thus we can define \(\mathbb{Q}_{b,d}\) as the array of equally spaced Rational numbers in base \(b\). With this array and the matrix of root of unity, which I explained about in Matrix Fast Fourier transform(MFFT) ²⁷, we can implement Imaginary numbers, \(\mathbb{C}_{b,d}\). In this way we have constructed the foundation for starting constructing Calculus and further advanced topics. At least we must agree on the accuracy first then we can reliably sum up two numbers with these definitions, not like two Real numbers that starting from digits with lower significant would potentially change the result of their sum!

Taking Constructivity's principles in Physics will give us the time and more! Let's try to answer questions that generally people think they are unanswerable with our knowledge!

What's the science?

First we need to be aware that construction is very similar to computation. In both of them you start from a foundation. In construction the foundation is the axioms, and in the computation they will be initial values. If we can express the axioms in numbers, like what Gödel constructed, then someone can argue there is no difference between them. On the other hand, the Constructivity argues observables, based on the principles, are constructable since reality computed them. Thus we can use them for constructing further to arrive to another observables. Since the reality did its own computation as well, we will both will reach to the same result, which is exactly what I have called Einstein's power. The result of our computation is called prediction, and the result of reality's computation is called experiments' measurements. They have been matched as far as human mind started to computing, and developed reasonably good models to calculate with. Hence, we call these models and computation the Science. As far as I know, nobody answered why the Science works, but here we go. It's an sign that the Constructivity is the only game in the town.

Why is the unreasonable effectiveness of mathematics?

This is a famous question ²⁸ that have been asked by many great minds. Here, the Constructivity has a clear answer for it. The reality is constructable, and good math is constructable, so why shouldn't they give different results?! It looks totally reasonable when you have Constructivity in your disposal. In fact, if you have Constructivity, it's unreasonable if the reality doesn't follow a good enough constructable model!

What is the time?

Yet another question that a lot of people asked but yet to be found a consistent answer is "what's the time?". The answer of Constructivity is the past already constructed, and the future is not, and nobody can get ahead of all the details of the reality's computation to make the future concrete. The boundary between these two is the present, and these three together make the time. Notice, this also implies the Causality if you ask!

I saw people's reaction to this argument, which mostly was: it's a contradiction with the Special and General Relativity, especially the block universe. Einstein himself found only the block universe consistent with the Special and General Relativity, so he argued the time is an illusion. Unfortunately, most people only have read in the books that there's no notion of now in those theories, so they would be surprised by my claim! However, what they call the notion of now in the text books is a \(3D\) plan, in the \((3+1)D\) space-time, that everyone agrees that's now. Here, we totally agree that there's no agreeable \(3D\) plan for the "now" for all observers/coordinates. But if we take the boundary between the past and the future as a \(3D\) jagged surface that's aligned with the null-geodesics, aka light cones, then there's no contradiction, since for every observer in any time the past is fixed, and the future is not-known equally. It's a totally constructable surface. Hope you like my spacetime diagram ²⁹ below! It shows the jagged surface that everyone agrees about where is the past and where is the future.

The nice thing about this jagged surface is that the far enough cones on that surface can be computed independently, which makes sense if the computer of the universe wants to do it multi-threaded! Here, I am not claiming there's a computer to compute the universe! The universe could be able to compute itself, so my point in software engineering's terms is only that the implementation of reality is modularized-on-runtime/isolated for big enough patches.

why do we have one direction for time?

It's constructable by using recursive functions, recall! Do I need to explain more? However, it's nice to think about possibility of time machines. If I create a time machine today to make a contradiction in the already computed past, I am referring to the Grandfather paradox ³⁰, then the computation needs to be repeated to make it consistent again. Unfortunately, constructing that algorithm is not viable for me now, but the end result must be a universe without the time machine, or a time machine that its crew has never interacted with the past!

This is good, since metrics, like Gödel metric ³¹, that has a closed timelike curves cannot be refuted by only Relativity, however, by adding Constructivity to the axioms we can safely reject them.

Does omnipotent god exist?

To just mention the range of answers the Constructivity can provide, it's nice to answer this one too.

If there was such a thing, even it couldn't avoid the cost of computation, like how a Platonist believes, since our real universe is a computation. If it could have reached to the final answers, it would have skipped this computation that we are living in! By the way, if you are a scientist, and you believe on a such god, you are in contradiction, and probably a lot of pain as the consequence.

Notice, this is the difference between classical determinism and the modern determinism the Constructivity dictates. The classical determinism say there's an omnipotent god knows the result of this computation, without providing an implementation for such an assumption! It's not possible based on the Constructivity's principles. In Constructivity, the cost of computation must be paid, which means the result of current ongoing computation is not known to any entity/agent! Be aware, the deterministc coordinates, that I mentioned before³², only have all the knowledge of the computed past, and in a lot of cases no conscious observer can measure in that coordinate. Therefore, Constructivity and existence of an omnipotent entity are in contradiction. Choose one!

Now that we talk about gods, and also power, it's nice to mention the connection in my brain, if you don't confuse it with the Constructivity's results! In my mind the Einstein's power is the highest power a god can achieve, so basically people like Einstein are the most powerful gods you can get. And guess what! By teaching the Constructivity I am teaching people to be gods! I can hear someone is saying: "Look babe! In the corner of the Internet someone is teaching people how to be gods!" By the way, I am teaching the way, but I am not a god, just like some of my colleagues in the universities who are teaching the way to be a scientist, but they are not scientists!

What's the consciousness?

The consciousness doesn't have an agreeable definition, so honestly, I couldn't find a measurable way to define it too! However, as far as I read the discussions about this topic, I noticed people expect a consciousness being to be responsible, which directly adds the property of freewill for that consciousness being. Even though personally, being responsible is not a constraint for me, but if people ask for it! Basically I am just arguing that I'll discuss freewill instead of consciousness here. Hope it's satisfying!

Unfortunately, as soon as we discuss the freewill, we need to discuss deterministic reality. It's kind of human nature to spontaneously ask those questions. Therefore, we have to address them. Fortunately we are ready to answer since we studied the Geometrical probability².

In one of the last posts ³² we argued that even if our model of reality is not deterministic, it must have a room for a coordinate system for a hypothesis observer who knows everything about the past of the system, which is the already computed information, so in the eyes of that observer the reality is deterministic. In other words, the model should provide all kind of coordinates and this is independent of answering if any conscious observer could measure with that coordinates. As argued, the reality as the computation is deterministic if your coordinate in the counting manifold ² implies you know enough.

Here, we kinda defined Consciousness based on freewill, and also, we know as a fact that brain is the conscious part of the body. Additionally, the brain is known to be chaotic, in the terms defined in Chaos theory³³. It's important to realize the chaos is deterministic, since a chaotic function, with the same inputs will compute the same results as before. Even though changing the inputs a little bit will change the output drastically different, also known as the butterfly effect ³⁴, which is the definition of a chaotic system. It's also worth noting that the Halting problem ³⁵, and the Computational irreducibility ³⁶ are features of the drastically different outputs of chaotic systems. This means slight changes in the past of a brain will result in very different future of that brain. This implies the coordinate system that brain observes in the Geometrical probability must always include probabilities and uncertainties. The brain cannot measure in a coordinate system that knows everything about itself, due to its chaotic structure.

These all conclude that for the brain's coordinate the inevitable probabilities gives room to observe its own freewill, even though another coordinate system must be outside the brain that measures the reality as a deterministic system. Therefore, in this topic I am totally on the Robert Sapolsky ³⁷ side, when he says "I was thinking this way since I was 14, and I can actually function this way about 3 minutes every other month". This is because he/we cannot get rid of chaotic behavior of our brain. By the way, his ideas are good until he also mentioned "punishment makes no sense", and he's right, but "consequence" makes sense in a deterministic system. Like when a ball hits a wall, it'll jump back. Someone can say it was the ball's punishment, since it hits the wall, but it was just the consequence of hitting the wall. It would be nice to change the language, but something like jail is a valid consequence of certain kind of behaviors, since it separates a potentially dangerous person from the society. Now go and try to define "a potentially dangerous person"!

Is the Constructivity the same as the Simulation hypothesis?

The Simulation hypothesis ³⁸ is suggesting the reality is simulated. And when you look at all the software simulators, none of them is Constructive, since they don't need to! They only need to recreate the part of the reality that is measured by the user. Basically, it's very similar to the Copenhagen interpretation ¹⁸, when you observer a wavefunction and it collapses. It's the opposite of being constructable! It's happening due to the fact that simulators need conscious observers, like users, and there's no point to compute anything more than those observes' measurements.

As mentioned, the Constructivity is following up the Relativities ¹³. Also, the first principle of Special Relativity ²¹ is clearly saying there's no difference between a conscious observer and other observes. Hence, simulators doesn't even satisfy the Relativity!

Another stronger argument against the Simulation hypothesis is that the parent universe of any simulated reality must have much more entropy, since errors are always there, or perfectly efficient computation is not achievable. The consideration for the amount of entropy, which is the capacity of a system to keep information, is showing every time a parent universe build a simulation, they sacrifice some of the entropy, which will stay outside the simulation, therefore, the simulated universe always has less, and less, entropy. By the way, based on our technology the amount of lost entropy that you cannot wire into the simulated universe is humongous. This makes it is very unlikely that we're in any simulation.

Especially if you check the models of reality by your own logic, and also, in everyday life you keep checking the mathematics of your surrounding, you probably noticed that you can trust it to correctly compute the reality, not just providing only your measurables. That's why it's all constructable.

Conclusion

The Geometrical probability can help us build probability without any dependency to the Set Theory.

There is a power, that I called Einstein's power, where someone can predict how reality MUST work. I devised the Constructivity with the following principles to have to enable this power for all of us!

• Construct everything on top of Natural numbers, arrays, and recursive functions ^{6 14}.
• Construction is a step-by-step evolution by using recursive functions.
• The reality, as well as the observations, aka the measurements, are all constructable.
• Additional to optimizing axioms to be minimum, the axioms MUST be observable.

Then after that We jumped on the deterministic boat and provided an explanation for why you can observe you are having freewill.

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References

Cite

If you found this work useful, please consider citing:

@misc{hadilq2025Constructivity,
    author = {{Hadi Lashkari Ghouchani}},
    note = {Published electronically at \url{https://hadilq.com/posts/constructivity/}},
    gitlab = {Gitlab source at \href{https://gitlab.com/hadilq/hadilq.gitlab.io/-/blob/main/content/posts/2025-05-06-constructivity/index.md}},
    title = {Constructivity},
    year={2025},
}