1. The understanding beyond formalism

To consider what computability is, the concept of a universal Turing machine should be invoked. A universal Turing machine can perform any computation; nevertheless, there is a thing that is considered to be out of the realm of what it can do. It can be proved that there does not exist a program that successfully judges whether a universal Turing machine halts or not, which means that for a given problem, there is no way to know if it is solvable or not. Although it has not yet been proved whether computation that our mind can carry out is beyond what a universal Turing machine can do, it should be noted that, at least, our mind can perceive that there is no way to know whether a given problem is solvable or not as if our mind exists outside the system that is concerned with the situation.
According to Godel's incompleteness theorem, within a mathematical system that is beyond a certain basic level of complexity, there exists a true mathematical statement that is not provable, also, there exists a consistent theory that cannot prove by itself its consistency. Nevertheless, at least, our mind can perceive that Godel’s incompleteness theorem is true as if we can look at the situation from the point of view that is outside the system, as similar to the argument of a universal Turing machine.
When considering only mathematical one, the truth, to which our mind can access, is something that goes beyond mere formalism.
Next, let us consider the phenomenon of understanding by Searle’s Chinese room argument. The Chinese room argument is as follows:
Imagine one is locked in a room, and although she does not understand Chinese, she is given a rulebook in English for manipulating Chinese symbols. The rule specifies the manipulation of the symbols purely formally, in terms of their syntax, not their semantics. Suppose some questions in Chinese symbols were put into the room. She manipulates these symbols in Chinese into symbols in English according to the rulebook, and she put out the answer in English symbols. Although she does not understand anything at all semantic property of these symbols that she manipulates, from the point of an outside observer, she behaves exactly as if she understands Chinese but all the same she does not understand a word of Chinese.
A mere manipulation of symbols does not evoke any meaning and does not represent any understanding. The criticism against the argument is that the whole system understands Chinese; to which Searle replies that the formal system cannot get the syntax to the semantics.

2. Language comprehension as a class

We consider here language comprehension. The Turing test is widely regarded as a legitimate test whether the intelligence of a computer is equivalent to the human or not. Let us consider a specific conversation between two humans A and B, and try to emulate the part played by B with a computer. It is always possible to program the computer in such a way that the response of the computer is indistinguishable from its human counterpart B. Of course, in order to do so, we need to know what A is going to say beforehand. We can do that as long as we are dealing with a specific example of conversation. This is cheating, because the difficulty of producing a program that passes the Turing test consists in the fact that human conversation is often unpredictable, and we cannot know beforehand what A is going to say.
The above argument demonstrates that the ability to comprehend language should be considered in terms of a class.

3. Non-physical aspect of our mind

To examine whether our mind can be fully explainable by physics only, a thought experiment devised by the philosopher Frank Jackson should be considered. Mary, a neuroscientist in the 23ed century, is the world’s leading expert on the brain processes responsible for color vision. However, Mary has lived her whole life in a black and white room and has never seen any other colors. She knows everything there is to know about physical processes in the brain -- its biology, structure and function. Nevertheless, there is something crucial about color vision that Mary cannot know: what it is like to experience a color such as blue. This thought experiment shows that some aspect of our mind is not fully explainable in terms of physics only.

4. Two languages

Language is an entity, which provides us with a certain limited worldview, through which we construct our worldview at least to some extent.
There are two languages, namely natural and mathematical ones, through which our mind perceives, depicts and understands many aspect of the world.
As far as our mind is concerned, the world is based on these two languages in one sense.
Everything, including the human brain, in the universe evolves in accordance with the laws of physics, whose temporal evolution is described by mathematical language. Since the process of natural language is a phenomenon in the brain, the process of natural language can be ultimately described by mathematical language and reduced to mathematical formalism. Therefore, the property of natural language, including its semantic aspect, derives from entirely mathematical formalism.
Can it be true? A radical statement should be made that natural language cannot be reduced to mathematical formalism; natural language exists independently on its own.

5. The universal process of natural language

The fact that children can acquire the ability of manipulating every natural language must show that the required process of every natural language is the same at the fundamental level, that is to say, at the unconscious level.
When one uses the term ‘natural language’, she should refer to that fundamental process of natural language at the unconscious level, specifically in the context of philosophy and the mind-brain problem, if not in the context of linguistics. In fact, it should be proposed that revealing the universal structure of all natural language is what present linguistics must pursue. (See the theory of universal grammar and mentalese.)
Although it may not be a relevant issue directly, let us introduce Einstein’ s letter to Hadamard, which is as follows:

The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities, which seem to serve, as elements of thought are certain signs and more or less clear images, which can be ‘voluntarily’ reproduced and combined. The above-mentioned elements are, in my case, of visual and some muscular type. Conventional words or other signs have to be sought for laboriously and can be reproduced at will.

6. Language, qualia, and intentionality

One of the mental phenomena that independently exists of physicalism is quale, whose very nature is that it is quite impossible to convey to others how one exactly feels when she experiences the phenomenon.
The phenomenon of understanding through natural language seems to be similar to the phenomena of qualia in the sense that it is quite impossible to communicate to others how one really understands through natural language, since the meaning of natural language is essentially a private one.
The similarity may suggest that the two phenomena may be closely connected in as yet unknown way.
Furthermore, a particular state of being, concerning our mind, which is being directed toward something, namely, “intentionality” might be the essential property of our cognitive faculty to endow the meaning of natural language in our mind.

7. Additional comment concerning language

Although our perception must be reduced to the neural correlates only, as far as our natural language processing is concerned, some interaction and synchronization with other subject’s process of the brain is required, as one of the facts concerning natural language shows that one must be exposed to the stimulus of language when she is in a particular age to acquire the language ability. This requirement of interaction and synchronization for natural language processing must be supported by the existence of mirror neurons at Broca’s area, which controls motory function for natural language (see also Libermann’s motor theory). Therefore, some sort of response selectivity, interaction and, synchronization are responsible for our ability of manipulating natural language. Furthermore, as the origin of language was mimicking the sounds and voices of nature and other animal with a gesture, it can be postulated that language is the product of the interaction with the environment and other subject’s process of the brain. Therefore, it can be postulated that the nature of language can be reduced to the relationship between one’s pattern of neural firing and other subject’s pattern of neural firing only.

Our speaking natural language is natural, no one realizes what she speaks and means when she manipulates natural language, the fundamental structure of natural language that enables us to communicate to others through natural language must be tremendously complex; in fact, it may be impossible to find the logical structure in our everyday language. Nevertheless, we may hope that some theory of computation and logic would provide us with the notion of what we can and cannot know about the world around us through language, and the notion might tell us something about the true nature of language.

Perception is a contemplation that is based upon two languages in the sense that without these media, there cannot be any meaning in our world as a human being who perceives the world, and also, in the sense that there cannot even be such a phenomenon as understanding at a first place. It follows that elucidating the nature of language would lead to the worthy proposal for epistemology.

When we discover the true nature of two languages, that are natural and mathematical ones respectively, the very nature is described by these languages themselves. Which means that we are eternally encapsulated in the world that is delineated by two languages; in other words, we cannot escape from the windowless, doorless tiny cell that is stated by two languages .

As mentioned at first in this section in a different fashion, unlike other mental phenomenon, the process of language is connected to the relationship between our mind and (for the lack of better term) external object. Language can be the product of the whole system.
Wittgenstein asserts that the very nature of language lies outside of where language itself exists; that is to say, language itself cannot speak anything about language.
Whereof one cannot speak, thereof one must be silent.
What is external object that enables us to speak about?

8. A comment on mathematical language

It is generally stated that mathematical language is more reliable medium to depict the world than natural language that is ambiguous and ill defined. However, as we cannot become able to understand mathematical language without the medium of natural language; to us, natural language is more dominant entity than mathematical language.

9. Plato's world

In respect of the understanding of mathematical truth, the mathematician Roger Penrose mentions as follows:

"I cannot help feeling that, with mathematics, the case of believing in some kind of ethereal, eternal existence"
"One's mind makes contact with Plato's world whenever it contemplates a mathematical truth"
"When one sees a mathematical truth, one's consciousness breaks through into this world of ideas, and makes direct contact with it.

As natural language has the same property with mathematical language in the respect that the both are the entities through which we perceive and understand the world, there is a possibility that the phenomenon of understanding through natural language might be connected to the property of Plato's world for its own.

10. Consciousness, qualia, and Plato's World

The neural firings in the brain are discrete and full of noise. Any causal relationship between neural firings is therefore expected to be discrete and full of noise. The quale that emerges in our mind, on the other hand, appears to possess what someone calls “features of the Platonic world”, namely qualities that seemingly reflect complete mathematical relationships. A certain quale is generated in our mind when a certain (ultimately mathematical) relation between neural firings is realized. We have qualia in our mind when we are conscious. Therefore, one of the functions of consciousness would be to interpolate the incomplete causal relationships between neural firings (which underlie our perception according to Mach’s principle in perception). In this picture, consciousness exists in order to map the discrete and incomplete causal relationships into the complete, mathematical relationships.

11. The limitation of our cognitive faculty

The world that we perceive, depict and understand through two languages is a certain limited reflection of the truth of Plato's world. How closely our perception is approximated to the structure of Plato's world is the limitation of cognitive faculty of our mind.

12. Philosophical approach toward the true nature of language

When the activities of Broca’s and Wernicke’s areas at the neural level become clear, would describing the model of natural language processing using those lead to some elucidation of the essential nature of natural language?
While we can learn much from the investigation of language as one of our cognitive processes in terms of brain science; at present when we cannot know what is going on concerning the process of natural language at the neural level, we need to approach philosophically toward the problem of language as we admit that our brain exists in the realm of where mathematical formalism governs and that our world is logical.

13. Conclusion

The phenomenon of understanding, perception and language are closely related and interconnected in as yet unknown way. When we seriously investigate the nature of these phenomena and also elucidate the relationships of them, we must go beyond formalism, functionalism, and physicalism; we must go, at least partially, toward metaphysics.

Appendix 1:
The overview of the incomputability on Turing machine

We number the Turing machines as T^n, and write the output of the nﾕth Turing machine when provided with an input of m as T^n(m), where n and m are natural numbers. There is a universal Turing machine, which can perform the computation of the nﾕth Turing machine if provided with the initial input of n. We do not consider here what significant difference the limitations on the physical memory space and computation time makes on the computational ability of the Turing machines. We can prove that there is at least one computation that a universal Turing machine cannot do, namely to determine whether the n'th Turing machine, when provided with the input m, will halt or not. A program that successfully judges whether a Turing machine halts or not is nothing else than a function h (n, m) which takes the value of either 0 or 1 in such a way that when h (n, m)=1 the computation halts, and when h (n, m)=0, it does not. It can be proved, using Cantor's diagonal slash argument, that there is no single Turing machine that can implement this particular function h. Therefore, there is at least one function (namely the function h) that a Turing machine cannot implement.

Appendix 2:
Mach’s principle in perception

Today, it is reasonably assumed that any aspect of mental phenomenon is ultimately explainable in terms of the neural firing in the brain. In 1972, Horace Barlow of Cambridge University applied the neuron doctrine to the problem of perception. Barlow suggested that the characteristic of our perception is specified by the nature of neural firings in our brain only. That is to say, no matter what mechanisms are involved, the properties of our mind is to be determined by the neural firing state only; a non-firing neuron is as good as non-existent as far as perception is concerned. The characteristics of our perception at one psychological moment should be explained by the nature of neural firings at that psychological moment only.
“Response selectivity” has been a concept of central importance in the discipline of perception. One idea that emerges is to assume that when a neuron with response selectivity to a particular external object fires, the perception of the object occurs. Unfortunately, there is a fundamental flaw in this line of argument, which becomes apparent when one tries to answer the following question. “When a neuron selectivity responsive to a feature A fires, how does the brain (or the subject) know that it is selectively responsive to feature A?” That is to say, response selectivity seems to assume that the relationship between external object and mental phenomenon has been already established, as “It’s already something out there in the environment.”
Ernst Mach was a physicist, philosopher, psychologist who had a great influence on Albert Einstein in his development of the theory of relativity. “Mach’s principle” states that the mass of a particle is determined by its relation to all the other particles in the universe. In general, the properties of an individual are determined by its relation to other individuals in the system.
A similar line of thought is relevant when we consider how our perception is formed through the neural firings in our brain. Namely, a neural firing plays a particular role in our perception, not because it is selectively responsive to a particular external object, but because the neural firing is related to other neural firings in the brain in such a way that the particular role in perception is endowed on the neural firing in question. The idea is “Mach’s principle in perception”.

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