Micro-level water-resources engineering—10: A bridge to end droughts?…

Let me ask you a simple question: Why are bridges at all necessary? I mean to refer to the bridges that get built on rivers. …Why do you at all have to build them?

Your possible answer might be this: Bridges are built on rivers primarily because there is water in the rivers, and the presence of the water body makes it impossible to continue driving across the river. Right? OK. Good.

In India, “kachchaa” (untarred) roads often exist on the sides of the main road or a high-way, as we approach a bridge on a river. These side-roads usually aren’t built after planning, but simply are a result of the tracks left by the bullock-carts plying through the fields, on both sides of the road. People from nearby villages often find such side roads very convenient for their purposes, including accessing the river. The sand-smugglers too find such approach-roads very convenient to their purposes. The same roads are also found convenient by journalists and NGO workers who wish to visit and photograph the same river-bed as it turns totally dry, for quite some time before summer even approaches.

Somewhere in there lies a certain contradiction—a technical contradiction, I should add.

If there were to be no water, ever, in these rivers, then no bridges would at all be necessary. Yet, these bridges are there. That’s because, in monsoon, it rains so much that these rivers begin to flow with full capacity; they even overflow and cause extensive flooding in the adjacent areas. So, naturally, bridges have to be built.

Yet, come even just late winter time, and the river-bed is already on its way to going completely dry. The bridge might as well not have been there.

Thus, the bridges, it would seem, are both necessary and not necessary in India. That’s the contradiction I was talking about.


But why not turn this entire situation to your advantage, and use the very site of a bridge for building a small check-dam?

After all, the very fact that there is a bridge means:

there is enough water flowing through that river, at least during monsoons. We only have to find a way to use it.


Here are some of the advantages of building check-dams nearby a bridge—or may be even directly underneath its span:

  • The patterns of water-flow across the pillars of the bridge, and even the pattern of flooding near the site of the bridge, has become well known, even if only because there is a better access to this site (as compared to other potential sites for a check-dam)—because of the existence of the main road.
  • There is already a built structure in place. This means that the nature of the rocks and of the soil at the site is already well studied. You don’t have to conduct costly geological surveys afresh; you only have to refer to the ready-made past reports.
  • Another implication of there being a pre-existing structure is this: The nearby land has already been acquired. There is no cost to be incurred in land acquisition, and the cost and other concerns in relocating the people.
  • Columns/pillars of the bridge already exist, and so, the cost of building the wall of a check-dam can come down at least a bit—especially if the wall is constructed right underneath the bridge.
  • Many times, there also is a lower-level cause-way, or an older and abandoned bridge lying nearby, which is no longer used. It can be dismantled so that the stones used in its construction can be recycled for building the wall of the check-dam. It’s another potential reduction in cost (including in the material transportation cost).
  • The existence of a bridge at a site can often mean that there is likely to be a significant population on either sides of the river—a population which had demanded that the bridge be built in the first place. Implication: If a water body comes to exist at this same site, then the water doesn’t have to be transported over long distances, because a definite demand would exist locally. Even if not, if the check-dam is equipped with gates, then the stored water can be supplied at distant locations downstream using the same river—you don’t have to build canals (starting from the acquisition of land for them, and further costs and concerns down the way).
  • Easy access to transportation would be good for side-businesses like fisheries, even for building recreational sites. (Think agro-tourism, boating, etc.)

Of course, there are certain important points of caution or concern, too. These must be considered in each individual case, on a case-to-case basis:

  • The local flow pattern would get adversely affected, which can prove to be dangerous for the bridge itself.
  • There is a likelihood of a greater flooding occurring in the nearby locations—esp. upstream! A blocked river swells easily, and does not drain as rapidly as it otherwise would—the causeway or the spillway can easily turn out to be too small, especially in the case of small dams or check-dams.
  • The height of the bridge itself may be good, but still, the river itself may turn out to be a little too shallow at a given location for a check-dam to become technically feasible, there. Given the importance of the evaporation losses, the site still may not turn out to be suitable for building a check-dam. (For evaporation losses, see my last post in this series [^].)

But overall, I think that the idea is attractive enough that it should be pursued very seriously, especially by students and faculty of engineering colleges.


We all know that there has been a great proliferation of engineering colleges all over the country. The growth is no longer limited to only big cities; many of them are situated in very rural areas too.

When a problem to be studied touches on the lives of people, say a student or two, it becomes easy for them to turn serious about it. Speaking from my own personal experience, I can say that BE project-reports from even relatively lower-quality engineering colleges have been surprisingly (unexpectedly) good, when two factors were present:

(i) When the project topic itself dealt with some issue which is close to the actual life of the students and the faculty, to their actual concerns.

For instance, consider the topic of studies of design of check-dams and farm-ponds, and their effectiveness.

During my stint as a professor, I have found that rural students consistently show (across batches) reporting of the actual data (i.e., not a copy-paste job).

In fact, even if they were not otherwise very bright academically, they did show unexpectedly better observation abilities. The observation tables in their reports would not fail to show the more rapidly falling water levels in check-dams. Invariably, they had backed the data in the tables with even photos of the almost dried up check-dams too.

Yes, the photos were often snapped unprofessionally—invariably, using their cell-phones. (Their parked bikes could be easily visible in the photos, but then, sometimes, also the Sun.) No, these rural students typically didn’t use the photo-quality glossy paper to take their printouts—which was very unlike the students from the big cities. The rural students typically had used only ordinary bond-paper even for taking color printouts of their photos (invariably using lower-resolution ink-jet printers).

But still, typically, the set of photos would unambiguously bring out the fact of multiple field visits they had made, per their teacher. The background shrubs showed seasonal variations, for instance; also the falling water levels, and the marks of the salt on the dam walls.

Invariably, the photos only corroborated—and not even once contradicted—the numbers or trends reported in their observation tables.

Gives me the hope that one relatively easy way to identify suitable bridges would be to rely on students like these.

(ii) The second factor (for good, reliable field studies) was: the presence of a teacher who guides the students right.

No, he doesn’t have to have a PhD, or even ME for that matter. But he has to know for himself, and pass on to his students, the value of the actual, direct and unadulterated observations, the value of pursuing a goal sincerely over a course of 6–8 months—and the fun one can have in doing that.


OK, a bit of a digression it all was. But the point to which I wanted to come, was academics, anyway.

I think academic institutions should take a lead in undertaking studies for feasibility of converting a bridge into a check-dam. Each academic team should pick up some actual location, and study it thoroughly from different viewpoints including (but not limited to):

  • CFD analysis for predicting the altered water-flow and flooding patterns (with the water flow possibly designed to occur over the main wall itself, i.e. without a side-weir), especially for a dam which is situated right under a bridge);
  • FEM analysis for strength and durability of the structures;
  • Total costs that will be incurred; total savings due to the site (near a bridge vs. far away from it at some location that is not easy to access); and overall cost–benefits analysis; etc.

The initiative for such studies could possibly begin from IITs or other premier engineering colleges, and then, via some research collaboration schemes, it could get spread over to other engineering colleges. Eventually, this kind of a research—a set of original studies—could come to take hold in the rural engineering colleges, too. … Hopefully.


Should the government agencies like PWD, Irrigation Dept., or “private,” American concerns like the Engineers India Limited, etc., get involved?

Here, I think that the above-mentioned academic teams certainly are going to benefit from interactions with certain select institutes like (speaking of Maharashtra) CDO Nasik, and CWPRS Pune.

However, when it comes PWD etc. proper, I do think that they operate rather in a direct project-execution mode, and not so much in a “speculative” research mode. Plus, their thinking still remains grooved in the older folds such as: either have multi-purpose large dams or have no dams at all!, etc.

But, yes, CWPRS Pune has simulation facilities (both with physical scale-models, and also via computational simulation methods), and CDO Nasik has not only design expertise but also data on all the bridges in the state. (CDO is the centralized design services organization that is responsible for engineering designs of all the dams, canals, bridges and similar structures built by the state government in Maharashtra.) The cooperation of these two organizations would therefore be important.


In the meanwhile, if you are not an engineering student or a faculty member, but still, if you are enthusiastic about this topic, then you can do one thing.

The next time you run into a site that fulfills the following criteria, go ahead, discuss it with people from the nearby villages, take a good set of snaps of the site from all sides, write a very small and informal description including the location details, and send it over by email to me. I will then see what best can be done to take it further. (The fact that there were so few engineering colleges in our times has one advantage: Many of the engineers today in responsible positions come from the COEP network.)

The absolutely essential criteria that your site should fulfill are the following two:

  1. The river gorge must be at least 25 feet deep at the candidate location.
  2. The under-side of the bridge-girder should itself be at least 35 feet above the ground or at a higher level (so that there is at least prima facie enough of a clearance for the flood water to safely pass through the bridge). But please note, this figure is purely my hunch, as of now. I may come back and revise this figure after discussing the matter with some researchers/IIT professors/experienced engineers. For visualization, remember: 10 feet means one storey, or the height of a passenger bus. Thus, the road should lie some 4 stories high from the river-bed. Only then can you overcome evaporation losses and also have enough clearance for flood water to safely pass through without doing any damage to the bridge or the dam.

Further, the preferred criteria (in site selection) would be these:

  1. The upstream of the site should not have too steep a gradient—else, the storage volume might turn out to be too small, or, severe flooding might occur upstream of the check-dam! For the same reason, avoid sites with water-falls nearby (within 1–2 km) upstream.
  2. The site should preferably be situated in a drought-prone region.
  3. Preferably, there should be an older, abandoned bridge of a much lower height (or a cause-way) parallel to a new bridge. Though not absolutely necessary I do include this factor in searches for the initial candidate locations, because it indirectly tells us that enough water flows through the river during the monsoons that the cause-way wouldn’t be enough (it would get submerged), and therefore, a proper bridge (which is tall enough) had to be built. This factor thus indirectly tells us that there is enough rainfall in the catchment area, so that the check-dam would sure get filled to its design capacity—that one wouldn’t have to do any detailed rainfall assessment for the catchment region and all.

So, if you can spot such a site, please do pursue it a bit further, and then, sure do drop me a line. I will at least look into what all can be done.


But, yes, in India, bridges do get built in the perennially drought-prone regions too. After all, when the monsoon arrives, there is flooding even in the drought-prone regions. It’s just that we haven’t applied enough engineering to convert the floods into useful volumes of stored water.

… For a pertinent example, see this YouTube video of a bridge getting washed away near Latur in the Marathwada region of Maharashtra, in September 2016 [^]. Yes, Latur is the same city where even drinking water had to be supplied using trains, starting from early April 2016 [^].

So, we supplied water by train to Latur in April 2016. But then, in September 2016 (i.e. the very next monsoon), their local rivers swelled so much, that an apparently well-built bridge got washed away in the floods. … Turns out that the caution I advised above, concerning simulating flooding, wasn’t out of place. …  But coming back to the drought-prone Latur, though I didn’t check it, I feel sure that come April 2017, and it was all back to a drought in Latur—once again. Fatigue!


PS: In fact, though this idea (of building check-dams near bridges) had occurred to me several years ago, I think I never wrote about it, primarily because I wasn’t sure whether it was practical enough to be deployed in relatively flatter region like Marathwada, where the drought is most acute, and suitable sites for dams, not so easy to come by. (See my earlier posts covering the Ujani and Jayakawadi dams.) However, as it so happened, I was somewhat surprised to find someone trying to advocate this idea within the government last year or so. … I vaguely remember the reports in the local Marathi newspapers in Pune, though I can’t off-hand give you the links.

On second thoughts, here are the links I found today, after googling for “check dams near bridges”. Here are a couple of the links this search throws up as of today: [^] and [^].

… Also, make sure to check the “images” tab produced by this Google search too. … As expected, the government agencies have been dumb enough to throw at least some money at at least a few shallow check-dams too (not good for storage due to evaporation losses) that were erected seemingly in the regions of hard rocks and all (generally, not so good for seepage and ground-water recharge either). As just one example, see here [^]. I am sure there are many, many other similar sites in many other states too. Government dumb-ness is government dumb-ness. It is not constrained by this government or that government. It is global in its reach—it’s even universal!

And that’s another reason why I insist on private initiative, and on involvement of local engineering college students and faculty members. They can be motivated when the matter is close to their concerns, their life, and so, with their involvement the results can turn out to be very beneficial. If nothing else, a project experience like this would help the students become better engineers—less wasteful ones. That too is such an enormous benefit that we could be even separately aiming for it. Here, it can come as a part of the same project.


Anyway, to close this post: Be on the lookout for good potential sites, and feel free to get in touch with me for further discussions on any technical aspects related to this issue. Take care, and bye for now…


A song I like:

(Hindi) “chori chori jab nazare mili…”
Lyrics: Rahat Indori
Music: Anu Malik
Singers: Kumar Sanu, Sanjeevani

[A song with a very fresh feel. Can’t believe it came from Anu Malik. (But, somehow, the usual plagiarism reporting sites don’t include this song! Is it really all that original? May be…)]

 

 

I need a [very well paying] job in data science. Now.

I need a very well paying job in data science. Now. In Pune, India.


 



Yes, I was visiting Kota for some official work when at the railway station of the [back then, a simple little] town, on a “whim” (borne out of a sense of curiosity, having heard the author’s name), I bought it. That was on 14th July 1987. The stamp of the A. H. Wheeler and Company (Rupa Publications), so well known to us all (including IITians and IIM graduates) back then, stand in a mute testimony for the same—the price, and the fact that this little book was imported by them. As to bearing testimony to the event, so does my signature, and the noting of the date. (I would ordinarily have no motivation to note a fake date, what do you say?) Also notable is the price of the book: Rs. 59/-. Bought out of Rs. 1800/- per month, if I remember those days right (and plain because I was an M. Tech. from (one of the then five) IITs. My juniors from my own UG college, COEP, would have had to start with a Rs. 1200/- or Rs. 1400/- package, and rise to my level in about 3 years, back then.)

Try to convince my the then back self that I would be jobless today.

No, really. Do that.

And see if I don’t call you names. Right here.

Americans!


A song I like:

(English, pop-song): “Another town, another train…”
Band (i.e. music, composition, lyrics, etc., to the best of my knowledge): ABBA

Bye for now.


And develop a habit to read—and understand—books. That’s important. As my example serves to illustrate the point. Whether I go jobful or jobless. It’s a good habit to cultivate.

But then, Americans have grown so insensitive to the authentic pains of others—including real works by others. The said attitude must reflect inwards too. The emphasis is on the word “authentic.” If a man doesn’t care for another honest, really very hard-working man in pain but spends his intellect and time in finding rationalizations to enhance his own prestige and money-obtaining powers, by the law of integrative mechanism of conscisousness that is the law of “karma,” the same thing must haunt him back—whether he be a Republican, or a Democrat. (Just a familiarity with the word “karma” is not enough to escape its bad—or good—effects. What matters are actions (“karma”s), ultimately. But given the fact that man has intellect, these are helped, not obscured, by it.)

Go, convince Americans to give me a good, well-paying job, in data science, and in Pune—the one that matches my one-sentence profile (mentioned here) and my temperament. As to the latter, simple it is, to put it in one sentence: “When the time calls for it, I am known to call a spade a spade.”

And, I can call Americans (and JPBTIs) exactly what they have earned.

But the more important paragraph was the second in this section. Starting from “But then, Americans have grown so insensitive to the authentic… .”

Instead of “pains,” you could even add a value / virtue. The statement would hold.

 

 

On whether A is not non-A

This post has its origin in a neat comment I received on my last post [^]; see the exchange starting here: [^].


The question is whether I accept that A is not non-A.

My answer is: No, I do not accept that, logically speaking, A is not non-A—not unless the context to accept this statement is understood clearly and unambiguously (and the best way to do that is to spell it out explicitly).

Another way to say the same thing is that I can accept that “A is not non-A,” but only after applying proper qualifications; I won’t accept it in an unqualified way.

Let me explain by considering various cases arising, using a simple example.


The Venn diagram:

Let’s begin by drawing a Venn diagram.

Draw a rectangle and call it the set R. Draw a circle completely contained in it, and call it the set A. You can’t put a round peg to fill a rectangular hole, so, the remaining area of the rectangle is not zero. Call the remaining area B. See the diagram below.

The Venn Diagram

Case 1: All sets are non-empty:

Assume that neither A nor B is empty. Using symbolic terms, we can say that:
A \neq \emptyset,
B \neq \emptyset, and
R \equiv A \cup B
where the symbol \emptyset denotes an empty set, and \equiv means “is defined as.”

We take R as the universal set—of this context. For example, R may represent, say the set of all the computers you own, with A denoting your laptops and B denoting your desktops.

I take the term “proper set” to mean a set that has at least one element or member in it, i.e., a set which is not empty.

Now, focus on A. Since the set A is a proper set, then it is meaningful to apply the negation- or complement-operator to it. [May be, I have given away my complete answer right here…] Denote the resulting set, the non-A, as A^{\complement }. Then, in symbolic terms:
A^{\complement } \equiv R \setminus A.
where the symbol \setminus denotes taking the complement of the second operand, in the context of the first operand (i.e., “subtracting” A from R). In our example,
A^{\complement } = B,
and so:
A^{\complement } \neq \emptyset.
Thus, here, A^{\complement } also is a proper (i.e. non-empty) set.

To conclude this part, the words “non-A”, when translated into symbolic terms, means A^{\complement }, and this set here is exactly the same as B.

To find the meaning of the phrase “not non-A,” I presume that it means applying the negation i.e. the complement operator to the set A^{\complement }.

It is possible to apply the complement operator because A ^{\complement } \neq \emptyset. Let us define the result of this operation as A^{\complement \complement}; note the two ^{\complement}s appearing in its name. The operation, in symbols becomes:
A^{\complement \complement} \equiv R \setminus A^{\complement} = R \setminus B = A.
Note that we could apply the complement operator to A and later on to A^{\complement} only because each was non-empty.

As the simple algebra of the above simple-minded example shows,
A = A^{\complement\complement},
which means, we have to accept, in this example, that A is not non-A.

Remarks on the Case 1:

However, note that we can accept the proposition only under the given assumptions.

In  particular, in arriving at it, we have applied the complement-operator twice. (i) First, we applied it to the “innermost” operand i.e. A, which gave us A^{\complement}. (ii) Then, we took this result, and applied the complement-operator to it once again, yielding A^{\complement\complement}. Thus, the operand for the second complement-operator was A^{\complement}.

Now, here is the rule:

Rule 1: We cannot meaningfully apply the complement-operator unless the operand set is proper (i.e. non-empty).

People probably make mistakes in deciding whether A is not non-A, because, probably, they informally (and properly) do take the “innermost” operand, viz. A, to be non-empty. But then, further down the line, they do not check whether the second operand, viz. A^{\complement} turns out to be empty or not.

Case 2: When the set A^{\complement} is empty:

The set A^{\complement} will be empty if B = \emptyset, which will happen if and only if A = R. Recall, R is defined to be the union of A and B.

So, every time there are two mutually exclusive and collectively exhaustive sets, if any one of them is made empty, you cannot doubly apply the negation or the complement operator to the other (nonempty) set.

Such a situation always occurs whenever the remaining set coincides with the universal set of a given context.

In attempting a double negation, if your first (or innermost) operand itself is a universal set, then you cannot apply the negation operator for the second time, because by Rule 1, the result of the first operator comes out as an empty set.


The nature of an empty set:

But why this rule that you can’t negate (or take the complement of) an empty set?

An empty set contains no element (or member). Since it is the elements which together impart identity to a set, an empty set has no identity of its own.

As an aside, some people think that all the usages of the phrase “empty set” refers to the one and the only set (in the entire universe, for all possible logical propositions involving sets). For instance, the empty set obtained by taking an intersection of dogs and cats, they say, is exactly the same empty set as the one obtained by taking an intersection of cars and bikes.

I reject this position. It seems to me to be Platonic in nature, and there is no reason to give Plato even an inch of the wedge-space in this Aristotlean universe of logic and reality.

As a clarification, notice, we are talking of the basic and universal logic here, not the implementation details of a programming language. A programming language may choose to point all the occurrences of the NULL string to the same memory location. This is merely an implementation choice to save on the limited computer memory. But it still makes no sense to say that all empty C-strings exist at the same memory location—but that’s what you end up having if you call an empty set the empty set. Which brings us to the next issue.

If an empty set has no identity of its own, if it has no elements, and hence no referents, then how come it can at all be defined? After all, a definition requires identity.

The answer is: Structurally speaking, an empty set acquires its meaning—its identity—“externally;” it has no “internally” generated identity.

The only identity applicable to an empty set is an abstract one which gets imparted to it externally; the purpose of this identity is to bring a logical closure (or logical completeness) to the primitive operations defined on sets.

For instance, intersection is an operator. To formally bring closure to the intersection operation, we have to acknowledge that it may operate over any combination of any operand sets, regardless of their natures. This range includes having to define the intersection operator for two sets that have no element in common. We abstractly define the result of such a case as an empty set. In this case, the meaning of the empty set refers not to a result set of a specific internal identity, but only to the operation and the disjoint nature the operands which together generated it, i.e., via a logical relation whose meaning is external to the contents of the empty set.

Inasmuch as an empty set necessarily includes a reference to an operation, it is a concept of method. Inasmuch as many combinations of various operations and operands can together give rise to numerous particular instances of an empty set, there cannot be a unique instance of it which is applicable in all contexts. In other words, an empty set is not a singleton; it is wrong to call it the empty set.

Since an empty set has no identity of its own, the notion cannot be applied in an existence-related (or ontic or metaphysical) sense. The only sense it has is in the methodological (or epistemic) sense.


Extending the meaning of operations on an empty set:

In a derivative sense, we may redefine (i.e. extend) our terms.

First, we observe that since an empty set lacks an identity of its own, the result of any operator applied to it cannot have any (internal) identity of its own. Then, equating these two lacks of existence-related identities (which is where the extension of the meaning occurs), we may say, even if only in a derivative or secondary sense, that

Rule 2: The result of an operator applied to an empty set again is another empty set.

Thus, if we now allow the complement-operator to operate also on an empty set (which, earlier, we did not allow), then the result would have to be another empty set.

Again, the meaning of this second empty set depends on the entirety of its generating context.

Case 3: When the non-empty set is the universal set:

For our particular example, assuming B = \emptyset and hence A = R, if we allow complement operator to be applied (in the extended sense) to A^{\complement}, then

A^{\complement\complement} \equiv R \setminus A^{\complement} = R \setminus (R \setminus A) = R \setminus B = R \setminus (\emptyset) = R = A.

Carefully note, in the above sequence, the place where the extended theory kicks in is at the expression: R \setminus (\emptyset).

We can apply the \setminus operator here only in an extended sense, not primary.

We could here perform this operation only because the left hand-side operand for the complement operator, viz., the set R here was a universal set. Any time you have a universal set on the left hand-side of a complement operator, there is no more any scope left for ambiguity. This state is irrespective of whether the operand on the right hand-side is a proper set or an empty set.

So, in this extended sense, feel free to say that A is not non-A, provided A is the universal set for a given context.


To recap:

The idea of an empty set acquires meaning only externally, i.e., only in reference to some other non-empty set(s). An empty set is thus only an abstract place-holder for the result of an operation applied to proper set(s), the operation being such that it yields no elements. It is a place-holder because it refers to the result of an operation; it is abstract, because this result has no element, hence no internally generated identity, hence no concrete meaning except in an abstract relation to that specific operation (including those specific operands). There is no “the” empty set; each empty set, despite being abstract, refers to a combination of an instance of proper set(s) and an instance of an operation giving rise to it.


Exercises:

E1: Draw a rectangle and put three non-overlapping circles completely contained in it. The circles respectively represent the three sets A, B, C, and the remaining portion of the rectangle represents the fourth set D. Assuming this Venn diagram, determine the meaning of the following expressions:

(i) R \setminus (B \cup C) (ii) R \setminus (B \cap C) (iii) R \setminus (A \cup B \cup C) (iv) R \setminus (A \cap B \cap C).

(v)–(viii) Repeat (i)–(iv) by substituting D in place of R.

(ix)–(xvi) Repeat (i)–(viii) if A and B partly overlap.

E2: Identify the nature of set theoretical relations implied by that simple rule of algebra which states that two negatives make a positive.


A bit philosophical, and a form better than “A is not non-A”:

When Aristotle said that “A is A,” and when Ayn Rand taught its proper meaning: “Existence is identity,” they referred to the concepts of “existence” and “identity.” Thus, they referred to the universals. Here, the word “universals” is to be taken in the sense of a conceptual abstraction.

If concepts—any concepts, not necessarily only the philosophical axioms—are to be represented in terms of the set theory, how can we proceed doing that?

(BTW, I reject the position that the set theory, even the so-called axiomatic set theory, is more fundamental than the philosophic abstractions.)

Before we address this issue of representation, understand that there are two ways in which we can specify a set: (i) by enumeration, i.e. by listing out all its (relatively concrete) members, and (ii) by rule, i.e. by specifying a definition (which may denote an infinity of concretes of a certain kind, within a certain range of measurements).

The virtue of the set theory is that it can be applied equally well to both finite sets and infinite sets.

The finite sets can always be completely specified via enumeration, at least in principle. On the other hand, infinite sets can never be completely specified via enumeration. (An infinite set is one that has an infinity of members or elements.)

A concept (any concept, whether of maths, or art, or engineering, or philosophy…) by definition stands for an infinity of concretes. Now, in the set theory, an infinity of concretes can be specified only using a rule.

Therefore, the only set-theoretic means capable of representing concepts in that theory is to specify their meaning via “rule” i.e. definition of the concept.

Now, consider for a moment a philosophical axiom such as the concept of “existence.” Since the only possible set-theoretic representation of a concept is as an infinite set, and since philosophical axiomatic concepts have no antecedents, no priors, the set-theoretic representation of the axiom of “existence” would necessarily be as a universal set.

We saw that the complement of a universal set is an empty set. This is a set-theoretic conclusion. Its broader-based, philosophic analog is: there are no contraries to axiomatic concepts.

For the reasons explained above, you may thus conclude, in the derivative sense, that:

“existence is not void”,

where “void” is taken as exactly synonymous to “non-existence”.

The proposition quoted in the last sentence is true.

However, as the set theory makes it clear and easy to understand, it does not mean that you can take this formulation for a definition of the concept of existence. The term “void” here has no independent existence; it can be defined only by a negation of existence itself.

You cannot locate the meaning of existence in reference to void, even if it is true that “existence is not void”.

Even if you use the terms in an extended sense and thereby do apply the “not” qualfier (in the set-theoretic representation, it would be an operator) to the void (to the empty set), for the above-mentioned reasons, you still cannot then read the term “is” to mean “is defined as,” or “is completely synonymous with.” Not just our philosophical knowledge but even its narrower set-theoretical representation is powerful enough that it doesn’t allow us doing so.

That’s why a better way to connect “existence” with “void” is to instead say:

“Existence is not just the absence of the void.”

The same principle applies to any concept, not just to the most fundamental philosophic axioms, so long as you are careful to delineate and delimit the context—and as we saw, the most crucial element here is the universal set. You can take a complement of an empty set only when the left hand-side operator is a universal set.

Let us consider a few concepts, and compare putting them in the two forms:

  • from “A is not non-A”
  • to “A is not the [just] absence [or negation] of non-A,” or, “A is much more than just a negation of the non-A”.

Consider the concept: focus. Following the first form, a statement we can formulate is:

“focus is not evasion.”

However, it does make much more sense to say that

“focus is not just an absence of evasion,” or that “focus is not limited to an anti-evasion process.”

Both these statements follow the second form. The first form, even if it is logically true, is not as illuminating as is the second.

Exercises:

Here are a few sentences formulated in the first form—i.e. in the form “A is not non-A” or something similar. Reformulate them into the second form—i.e. in the form such as: “A is not just an absence or negation of non-A” or “A is much better than or much more than just a complement or negation of non-A”. (Note: SPPU means the Savitribai Phule Pune University):

  • Engineers are not mathematicians
  • C++ programmers are not kids
  • IISc Bangalore is not SPPU
  • IIT Madras is not SPPU
  • IIT Kanpur is not SPPU
  • IIT Bombay is not SPPU
  • The University of Mumbai is not SPPU
  • The Shivaji University is not SPPU

[Lest someone from SPPU choose for his examples the statements “Mechanical Engg. is not Metallurgy” and “Metallurgy is not Mechanical Engg.,” we would suggest him another exercise, one which would be better suited to the universal set of all his intellectual means. The exercise involves operations mostly on the finite sets alone. We would ask him to verify (and not to find out in the first place) whether the finite set (specified with an indicative enumeration) consisting of {CFD, Fluid Mechanics, Heat Transfer, Thermodynamics, Strength of Materials, FEM, Stress Analysis, NDT, Failure Analysis,…} represents an intersection of Mechanical Engg and Metallurgy or not.]

 


A Song I Like:

[I had run this song way back in 2011, but now want to run it again.]

(Hindi) “are nahin nahin nahin nahin, nahin nahin, koee tumasaa hanseen…”
Singers: Kishore Kumar, Asha Bhosale
Music: Rajesh Roshan
Lyrics: Anand Bakshi

[But I won’t disappoint you. Here is another song I like and one I haven’t run so far.]

(Hindi) “baaghon mein bahaar hain…”
Music: S. D. Burman [but it sounds so much like R.D., too!]
Singers: Mohamad Rafi, Lata Mangeshkar
Lyrics: Anand Bakshi

[Exercise, again!: For each song, whenever a no’s-containing line comes up, count the number of no’s in it. Then figure out whether the rule that double negatives cancel out applies or not. Why or why not?]


 

[Mostly done. Done editing now (right on 2016.10.22). Drop me a line if something isn’t clear—logic is a difficult topic to write on.]

[E&OE]