non-similar structure

Consider a microphone picking up sound and converting it to an electric signal. This is an instance of abstraction. The being asked, whether mathematics (an abstraction) is similar in structure to the "external world", can have meaning only if we can similarly ask whether the electrical signal generated by a microphone is "similar in structure" to the sound it is abstracting from.

How can we tell?

In fact we cannot. We can pass the signal through another device - a speaker and convert it back to sound, and then we can, with our ears, compare the reproduced sound in one ear to the original sound in the other ear - (but only by parallel abstractions into neurological inputs - which, being the same kind, the brain is capable of comparing).

We can also take ANOTHER microphone and connect its output, and the output of the first microphone - now both electrical signals - to an electric circuit designed for comparing two electric signals, and make a determination that these two parallel abstractions are similar to each other. But there is no possible way to compare two different levels of abstraction unless they are represented in the same medium (or at least in comparable mediums).

One can compare neurological inputs to each other and evaluate their similarity. One can compare verbal inputs (only by means of parallel transductions into neurological signals) to each other. One can compare the neurological abstraction from some verbal inputs to the neurological abstractions from some non-verbal inputs, but one CANNOT compare neurological inputs to any non-neurological levels of abstraction, just like it is not possible to directly compare electric signals to sound waves. They are at different levels of abstraction and in media of different types.

We can compare our neurological response to our exposure to mathematics with our neurological response to the "external world", but we cannot compare mathematics to the external world. They are different types of media at different levels of abstraction.

It is "received doctrine" within general semantics that Korzybski claimed that mathematics is the only language which is "similar in structure" to the external world.  I wrote a criticism of "similarity of structure" suggesting that the notion is either based on infinite regress or is in fact based on identity.

Close examination of the process of abstraction and the limitations on comparing things - which must be of the same type - shows that it is not possible to examine or test Korzybski's claim, and it is a claim that is not consistent with the notion that all knowledge is obtained via the process of abstracting from various sources - including memory.

What can it mean for a structure at one level of abstraction to be similar to a structure at another level of abstraction? We can only judge or evaluate this question based on performing a further abstraction from each of the two structures to be compared into a common third level of abstraction and then by comparing the resulting abstractions. Who knows what happened in the respective abstracting processes? What transformations happened? What was left out? There is no way to know, because the evaluation can only be done by more abstractions.

We are comparing something else when we evaluate that they are "similar". We are certainly not comparing the original different things to each other. To suggest that we are is to "identify" that which gave rise to the abstraction with the result of the abstraction.

So, to greatly annoy many general semantics who uncritically accept received doctrine, I say this:

Korzybski's claim is completely untestable. As such it is outside the purview of general semantics.

We cannot compare structures at different levels of abstraction unless they are represented in the same type of medium. And we actually do compare things only by performing highly individualized (idiosyncratic) parallel abstractions into the same neurological medium - individual brains. We cannot know the changes involved in any of these transformation-abstractions, so we cannot really know of any structures "in the external world". All we have is our highly individualized neurological models, which we further abstract into verbal levels.

To get from your brain to mine involves two highly individualized abstraction process. Going back involves two more highly individualized abstraction process.

So, for you to evaluate my response to your words, you see the result only after two highly individualized abstraction processes, some cogitation, and two more highly individualized abstraction processes.

  1. Your brain -(1)-> speaks words -> (noisy environment) -->> hears words -(2)->> my brain (thinks) -(3)->>> speaks words -->>> (noisy environment) -->>>> hears words -(4)->>>> your brain again.
  2. Your brain -(1)->>>>> your memory -(time delay) ->> your memory -(2)->>>>>> your brain.
  3. Your brain now (retrieved from memory) is compared to "your brain again (from 1)".

Now you can compare the result of two abstraction processes to the result of 4 abstraction processes and decide if what you understood by your inference as to what I meant by what I said as a result of what I inferred you meant from what I heard to your memory of what you said.

All this complexity is lost by many levels of identifying. How often do we say "But YOU SAID...", when we can really only testify to what we remember that we heard.

I cannot compare what you mean to what I mean. I can only compare what I mean to what I infer you intend based upon my abstractions from what I heard of your abstractions to represent what you mean.

So, it is very frequently the case that we cannot even compare things in the same type of medium. They must be in the same medium at the same level of abstraction - and then only by the process of comparing our individual abstractions.

So, my point, redundantly made, is that it is not possible to compare the structure of mathematics to the structure of the external world - period. We can each make our own abstractions from both and then say that our abstractions are similar, but that's all.


I chose a microphone for a couple of reasons. We would normally think that there is "really" a "faithful reproduction" from the input to the output with as little loss as possible. Technically this is simply false. The sensor has a limited range of frequencies to which it responds as well as a limited range of amplitude or energy that it can pick up. It can miss weak signals entirely and it can be "over-driven" and lose strong signals. It cannot pick up ultra ultrasonic or high frequencies and it cannot pickup subsonic or low frequencies. It is indeed an abstractive process. Our nervous systems abstract from the environment into neurological levels in an analogous fashion, but with many more levels of loss and consolidations. The subsequent abstraction into "emitted verbalizations" is even more restricting.

We "believe" that the oscillations we can see on an oscilloscope (electrical) are, in some way, "true" representations of the oscillations in the input sound. But if you look at the pattern of waves on the surface of water and also look at the oscillations displayed on the oscilloscope from the hydrophone (water microphone) we can see that the "structure" looks very different. (Since water waves are much more analogous to sound waves, and are slow enough to see, we have a reasonable approximation that can be viewed with the same sense - sight.)

The difference in structure between two levels of abstraction in such a physical device being so obvious shows that the difference in structure between "physical" and neurological responses, and between neurological and verbal responses, which are clearly much greater abstractions, won't be "similar in structure".

Because the "amount" of abstraction is "believed" to be so little in the processing of a microphone, the illustration that the difference is very large there sheds light on the differences there must be in human abstracting process and questions strongly our "assumption" of similarity of structure in abstracting.

What you say must be incorrect, as is shown by _endless_ examples in daily life. For instance, I may be _told_ that down at the zoo I will see lions, tigers, wolves, giraffes, antelope, etc., and lo and behold I go down to the zoo and what do I _see_ but Lion1, Tiger1, Wolf1, Giraffe1, Antelope1, etc. Now I have compared what I _saw_ with what I was _told_, and they _were_ similar in structure, despite being on different orders of abstractions, even though you may go to your grave denying, on verbal grounds, that this could possibly have happened.

In all these examples you have performed parallel abstractions from two different sources into your neurological process. It this in the medium of the brain of the observer where the actual comparison is being done. Never is it the case that one can compare what one was told (the auditory stimulus) to what one sees (the visual stimulus). One can ONLY compare the results of parallel abstraction from each of those experiences. You are in fact comparing nervous responses (abstractions from auditory stimuli) to nervous responses (abstractions from visual stimuli), and both of the structures you are comparing are the result of abstraction processes from DIFFERENT inputs (and quite different abstraction processes at that).

You have IDENTIFIED your inputs with your responses. All the "_endless_ examples in daily life" fail in the same way. It took millennia for us to get to consciousness of abstracting, but you throw it all away in one simple statement.

It is "received doctrine" within general semantics that Korzybski claimed that mathematics is the only language which is "similar in structure" to the external world. I wrote a criticism of "similarity of structure"  suggesting that the notion is either based on infinite regress or is in fact based on identity. 

You conflate "similarity" and "sameness" with "identity in 'all' respects", which is INCORRECT, and furthermore you use only incomplete or mistaken statements of the non-A premise, which in its complete and correct form says that "A map is not the territory that it represents". If you want to criticize gs by speculating on the wordings of incomplete paraphrases, or the formulations of incompetents, or the like, don't expect anything but incorrect results.

Not at all. I have consistently provided explanations of the differences between similar, same and identity, which is more than I can say for others. "Same" I take as a primitive yes/no evaluation in simple ordinary contexts illustrated by some example sentences. "Similar" I describe in terms of structure, relations, and degree of correspondence between parts such that the breaking down of parts into parts must stop at some point - recursively - with the bottom ending in the primitive yes/no same. Identity has two forms - the mathematical definition - which I use, and the trivial case of "identical" - "same in all respects" (We cannot define identical as "identical in all respects", because that is circular.) "Same in all respects" presumes the structure I speak of that can be broken down in to "respects", each of which can be compared (more below).

In abstracting, the map is not the territory. In every instance of the smallest and most minute step in abstracting, the lower level, the stimulus, is the territory, and the higher level, the response, is the map. Never is the map the territory; never is the stimulus the response; never is the higher level the lower level; never is the map the territory.

What do we mean by "similarity of structure"? One phrase that has sometimes been used is having some "invariance under transformation". In fact, from the mathematical source of this phrasing, it means that there is some measure in the domain of the function which is identical to the same measure in the range of the function. The general concept in this notion is that both the range and domain can be subdivided or broken down into parts such that there are some parts in the range that are the same as some parts in the domain - the part that is invariant - and there are some parts in the range that are not the same as some parts in the domain - the region where invariance does not hold. In this characterization, each of the range and domain are presumed to have some "structure", and what is meant by that is that they can each be divided into sub-structures which have relations among them.

For "invariance under transformation" to have any meaning, there must be a domain, a range, and a specified method of taking some subordinate element of the domain and mapping it to a subordinate element in the range. Moreover, the domain and range must be of the same type so that an element in the domain can be compared directly to an element in the range, because 'invariant' means "no change".

Many functions from one type of domain to another are possible, but none of these can have "invariance under transformation" unless it is possible for the mapping to produce exactly the same output as it had for input. Implicit in this notion is a particular characterization for "structure" - namely that it be capable of being broken down into sub-structures that are related to each other. Moreover, implicit in this notion is the "identity" of certain elements in the range and the domain - namely those that "are invariant" or do not change under the transformation (mapping).

Korzybski defines structure (S&S, 4th ed. p. 56) as "a complex of ordered and interrelated parts". "Two relations are said to be similar if there is a one-to-one correspondence between the terms of their fields, .... When two relations are similar, we say that they have a similar structure."

Structures are similar - according to Korzybski - means they can each be broken down into ordered and interrelated parts, and there is a one-to-one correspondence between the relations among their parts.

That agrees with "normal" definitions as well as that in my paper.

What we normally mean by "similar in structure" is that there are corresponding sub-structures and corresponding relations between two different structures. For this to happen, each structure must have sub-structure and relations that can be corresponded. Moreover, they must also have sub-structure that cannot be corresponded - so as not to be the same as each other.

 ... we cannot really know of any structures "in the external world".

What does it mean, to "really know"? Again and again, so many individuals who "disagree" with me fall back on the premise that we cannot "really know" this or that, but not once have any of them given me what it "is" to "really know".  --

The "normal" use of 'know' is based upon a correspondence theory of truth and an implicit naive realism metaphysics. A statement is true if and only if it corresponds to structures in reality. This presumes that things, structures, processes, etc., "exist" in the world independent of any observers. It also naively presumes that it is possible for words to correspond directly to these presumed to exist things. Persons "know" propositions - statements about the world - to be true just in case the statements do correspond to the world.

(Little of this "normal" usage is consistent with general semantics, and Korzybski often refers to Tarski, who first explicitly provided the mathematical model for the correspondence theory of truth - which is then extended to produce the many valued logics.)

The strong sense of 'to know' is that someone "knows" something if and only if it is true in the correspondence sense.

A weaker sense sometimes used by some people would correspond to what we normally use the term 'believe' for. The person has a very strong subjective feel that what it is that they "know" in this way is true.

"Really know" simply emphasizes the strong version.

I see no evidence that general semanticists have offered any other defining characterizations for the term 'know', and until someone provides as EXPLICIT FORMULATION or a "general semantics" definition for the term 'know', I have no alternative but to presume they mean something like the "normal" usage.

I don't know what you "normally" evaluate but most of us are quite used to telephones distorting voices, and plainly _do not_ evaluate that the microphone in it provides a completely faithful reproduction. Nevertheless we are willing to wager our lives, sometimes, on the similar in structure between what we say and what the person on the other end understands. Otherwise the telephone wouldn't bring the police or ambulance or fire engine.

This sort of thing happens again and again with "philosophical" methods -- some assertion is made that is plainly false in general, but the writer so puts it as to persuade the reader to imagine a limited situation in which the assertion is more or less correct and to imagine _only_ that situation. You want us to imagine a hi-fi stereo reproducing Beethoven rather than a cheap telephone bringing Walker Texas Ranger to our rescue, because there is a different "emotional" impact on your argument depending on what we imagine.

Thus (after some other stuff) you write:

Because the "amount" of abstraction is "believed" to be so little in the processing of a microphone, the illustration that the difference is very large there sheds light on the differences there must be in human abstracting process and questions strongly our "assumption" of similarity of structure in abstracting.

But no such "light" is shed if one imagines the pathetic microphone in the telephone that succeeds in bringing the police, despite all "philosophical" appeals to the general failure and ignorance of even the best abstractions.


Similarity "attributed" by the evaluator AFTER abstraction into a common medium - the brain; similarity attributed only because the inputs were IDENTIFIED with the responses.

Nothing has been said regarding whether or not the models were not disconfirmed, as would be the case when a call to 911 actually brought an ambulance. The veracity of the model is not an issue and is irrelevant - a red herring.

Methinks: there _must_ be similarity in structure, _somewhere_ in the electrical signals in the microphone etc. so long as we are to get a sound output similar to the sound input--

any dissimilarity possibly introduced would, rather expectedly, result in distorted output.

I dare think - a 'high-fidelity' transmission or reproduction of sound would itself be a sufficient proof of there going some sort of 'similarity of structures' along the wires.


We can evaluate that there is similarity between the original sound and the reproduced sound because we can hear that they sound similar - and this evaluation uses the same medium. But, the electrical structure can't be compared to the sound structure, so any assertion that there is "similarity of structure" between these two very dissimilar things is just a guess. The high fidelity output is only proof of "reproducibility" of something like the original, not that there is any similarity to intermediate processes.

The "usefulness" of the theory is in helping us understand consciousness of abstraction and aid us in not making unwarranted assumptions. If we are aware that whatever structure there is at the higher (map) level of abstraction is different from the structure at the lower (territory) level of abstraction even though there may be a transformation to convert one to the other we will be less likely to be blocked by assumptions about the nature of the presumed structure at the level of the territory.

Annotated bibliography of general semantics papers
General Semantics and Related Topics

This page was updated by Ralph Kenyon on 2009/11/16 at 00:27 and has been accessed 7473 times at 40 hits per month.