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In The Theoretical Minimum, in lecture 1, Leonard Susskind says that you can only visualize 3 dimentional images. (see yourself). Therefore, he says, in order to deal with N dimensions, you need to deal with abstract algebra. I wonder why the brain is hardwired for abstract algebra.

It might be worth to note that mnemotechnicians remind us that your mind thinks in pictures, and remembers pictures, not words or abstract thoughts?

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  • $\begingroup$ Here is the opinion of an advanced mathematics student. I don't believe the brain is "hardwired" for abstract mathematics (note that "abstract algebra" is actually a specific branch of mathematics). Perhaps it is more capable of abstract thinking in general, compared to other animals. For many, learning math requires arduous hardship, and even for those to whom math comes easily, intuition seems to universally be something developed and honed through experience rather than something already had. One might ask why the human brain has such hardware potential, I guess. $\endgroup$
    – anon
    Jul 14, 2013 at 19:06
  • $\begingroup$ @anon Yes. Your answer lacks the contrast with the visualization capacity. For me, the intuition is something like visualization. Intially, the picture of studied subject is blank. But, any subject, if it is not a mess, it has a structure. As you gain intuition, the picture starts appearing. You start seeing some contours of objects, their relations and ways to pass from this point to that point, similarly to orientation in 3d space with only exception that in abstract subject you have something like graph instead of 3d sweep. So, I wondering: what is the difference with visualization? $\endgroup$
    – Val
    Jul 15, 2013 at 7:25

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Abstract algebra and n-dimensional space are constructs of the human mind. Since they are constructs of the human mind, it is of course possible for the human mind to grasp them.

N-dimensional space is a mathematical construct. That human mathematics allows for the construction of something, does not mean that it exists in reality.

For example, in mathematics, there are 3 apples. In reality, there are no 3 apples, because all apples are different (in size, color, ripeness, taste etc.), and the equation 1 apple = 1 apple is always false.

Mathematics is an abstraction from reality, and the fact that it is an abstraction means that relevant information is lost in mathematical representation. If you turn this around, it means that a mathematical model has an unkown relation to reality, because you don't know which information is not present in the model.

Even more basic, it is unknown if anything relating to that model even exists. For example, the equation 3 - 4 = -1 has no relation in reality. There are no negative objects or lengths. (All scales that include negatives, such as temperature, have an arbitrary zero point and can be transformed to a scale without negatives, e.g. Kelvin.) Even simple mathematics is fictional. The existence of a mathematical object or model, such as n-dimensional space, does not prove, nor even imply, the existence of a related reality.

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  • $\begingroup$ I thought that human mind is hardwired to abstract. Abstraction (working with variables that cover infinities of particular cases instead of drilling every particular case) is the only means to handle complexity and making predictions, which favors survival. All apples are equal because we count the common property of every apple. But, I am not sure how it is related with vision. $\endgroup$
    – Val
    Jun 22, 2013 at 13:12
  • $\begingroup$ Yes, but you still can't deduce the real apple from the number 1. So how would deduce any reality from n-dimensional space? This abstraction is so far removed from a reality that we cannot even perceive, that it tells us nothing. Just as the 1 can stand for one apple, one human being or one act of murder, n-dimensional space can stand for anything -- or nothing. $\endgroup$
    – user3116
    Jun 22, 2013 at 14:15
  • $\begingroup$ I'm not sure how abstraction in the sense of grouping experiences into categories for the ease of orientation in reality relates to abstract algebra (which deals with algebraic structures, and not the categorisation of empirical reality). $\endgroup$
    – user3116
    Jun 22, 2013 at 14:22
  • $\begingroup$ I ask why you can see the abstract ideas but not N-dimensional space. I do not say that you should deduce the image of N-dimensional space from the abstract description of it. I care only why you can grasp one but not the other. If grain is flexible to learn whatever it should be able to start visualizing non-3d spaces. You was born without the capability to process abstract algebra. You developed this capacity. Why you cannot develop the capacity to see 1- or 4-dimentional lines? $\endgroup$
    – Val
    Jun 24, 2013 at 17:10
  • $\begingroup$ I'm not sure anyone really "sees" (in his mind's eye) abstract ideas. I certainly cannot. I find I always need to understand what mathematics does in relation to my physical and sensory reality, to comprehend it. Pure abstraction is beyond me. And I have an IQ of 140, so I would assume that a large proportion of the population has similar difficulties. Which makes your assumption that "our" (i.e. all human) brains are "hardwired" for abstraction somewhat dubitable. What mathematically talented people do with maths in their brains, I have no idea, but I would assume that it is not visual. $\endgroup$
    – user3116
    Jun 25, 2013 at 6:57
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One could say that brain is not hardwired for abstract ideas... why? There are exhaustive work by Piaget in which he claimed about stages of thinking. Modern researches argue that only 25-35% of High-school students dont entry Formal operational stage which is approximately abstract thinking. So average Joe is hardwired for 3 dimensional vision and concrete thinking.

update1: From theoretical perspective of evolutionary psychologist or biologist, it could be assumed that in one moment on earth it was more important for organism how to organize themselves, or how to find food etc then how to see in more dimensions. There is obvious difference in various species and vision, we can assume that it was developed chronologically or at once. but if it was chronologically the adaptation motive was the main motive. In that manner one of definition of inteligence is ability to adapt.

update2: i watched this video lecture and there are several errors in his statements: 1)time is perception of human, it is question if it is something which is universaly acknowledged. 2) 1, 2, 3 are not numbers - they are Plato representation of numbers which we adopt 3) this is more important 3 dimensional visionary system is like time, something we belive exists, but n-dimensional system and 3 dimensional system have only word 'dimension' in common. they are not same 'dimensions' you think it is the same because the name is same. Simmilar mistake is between Jungian word Extraversion and Eysenck or B5 term. Do you understand now? So people want to put in coordinate system something what does not belong there.

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  • $\begingroup$ Ok. I just cannot understand why people, who are both hardwired neither for abstract thinking nor n-dim visualization, can do the first but not the second, according to Susskind? How susskind be right? Is he right? Why are we slaves of our neural architecture w.r.t visulalization but not wrt abstract thinking? Why we can penetrate the abstract thinking barrier but not the n-dim visualization? $\endgroup$
    – Val
    Jul 10, 2013 at 13:17
  • $\begingroup$ i tried to answer that in second part of sentence. is there any evidence that there is more then 3d in real world? is there any organism (except God and Angels) which have vision of 3d? $\endgroup$
    – ICanFeelIt
    Jul 10, 2013 at 13:21
  • $\begingroup$ Adaptability just explains that you can learn both abstract thinking and visualization. It does not explain why you can learn first but not the other. $\endgroup$
    – Val
    Jul 10, 2013 at 13:22
  • $\begingroup$ I will have exactly the same argument in argument with evolutionist. But they assume that you have abilities which help species survive. So first it was vision which adapt and after that abstract thinking. you have species with dolphins which have 3d sound systems, but im not aware of 4d sensory system. $\endgroup$
    – ICanFeelIt
    Jul 10, 2013 at 13:41
  • $\begingroup$ Why students can develop the abstract thinking but not 1d sensory system? Why do I need 1d sensory system for visualizing 1d space? Which kind of sensory system is needed for abstract thinking? $\endgroup$
    – Val
    Jul 10, 2013 at 14:01

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