Plato extended

English: The School of Athens (detail). Fresco...

Plato conceived the idea that each and every real physical object is an inferior representation of an idealised “Form”. Forms were supposed to be “more real” in some sense than the ordinary real world objects that the Form idealises. Wikipedia says this, in its article on the theory of Forms:

These Forms are the essences of various objects: they are that without which a thing would not be the kind of thing it is. For example, there are countless tables in the world but the Form of tableness is at the core; it is the essence of all of them.

There are several issues with this approach. For instance, when one looks at a table, one can count its legs. Many tables have four legs, while some may have more than four and some may have less than four. Some may rest on a pedestal and so have no legs at all.

So what is the essence of a table? It seems that there are many attributes of tables that are variable, like the number of legs, which is pretty shoddy for an ideal of an object. One could define a table by the uses to which it is put, but again the uses to which a table can be put are variable. We have writing tables, operating tables and pool tables. It seems that there is no attribute can be found which has a single value which makes a table a table.

For Plato the Form of a physical object is the richer of the the two. In the Analogy of the Cave, the real world objects were mere shadows of the Forms. However, it appears that the real world objects have to be richer than the ideals of the objects. This derives from the necessity of distinguishing one real object from another. This table here is different from that table there (in spacial location if in nothing else), and specific location is an attribute that the ideal cannot have.

like being in the Plato’s cave
A shadow in Plato’s cave

Computer scientists have a solution to these issues, based around the concept of an “object”. An object is a container which may contain other objects, properties, or actions (called “methods” in the jargon) and which belongs to a class of objects. A table object for example may contain zero or several leg objects, it may contain the property of being wooden, and may contain the actions or methods of “lay”, “dine off”, “clear”, and so on.

English: Showing the main components of a class

Included objects can be considered parts of the object like the legs of the table. The properties are descriptive of the object, such as “wooden” or the property of “has four legs”. The actions or methods are things that you can do to the table, such as “dine off” it or “lay it for dinner”.

All objects belong to a class of objects and are called “instances” of the class. Each class is unique. There is only one class called “table” for example, and all table objects (instances of the class) derive or in other words are instantiated from it.  A class is also a container and may bring properties and methods to the derived instance, so the table class may contain an action or method of “lay for dinner” and a property of “has four legs”.

English: Diagram of relationship between objec...
English: Diagram of relationship between objects and classes (Photo credit: Wikipedia)

So far so good. The Form equates to the class and the real world object to the instance object, so the computer science model aligns with Plato’s model. However the computer science model has a few more wrinkles which Plato’s model doesn’t.

Firstly, when the object is instantiated in the computer science model, the new object doesn’t have to incorporate the properties and actions or methods of the class that it is modelled on. For example, if the new table is used for writing letters, then it doesn’t need the action or method of “lay for dinner” so it can omit it. On the other hand it may benefit from a “clear space” action or method! It would probably carry over a “has four legs” property if the class supplies it. The instantiation process is very flexible! Class properties and actions or methods can be included in the instance or new properties and actions or methods can be created, if required.

Secondly, each and every object can implement one or more classes. This is a fancy way of saying that it can pick and choose properties and actions or methods from one or more classes. In other words a table object may be made of wood, hence, to use the jargon, it inherits from the class of “wooden objects” the action or method “polish”. We can polish a wooden object, and we can eat off a table object, so we can do both off a wooden table.

English: answer to the question: Draw an inher...
Inheritance diagram

The point of all this is to address some of the deficiencies of Plato’s theory of Forms. Plato’s idea was that any table object derived from an archetypal table Form, which was not a real world object but still existed in some sense. The idealised Form was considered “purer” and was the essence of a table, comprising only those attributes shared by all tables. The difficulty here is finding any attributes that every table has, and which no “non-table” has. Using the extended Platonian scheme, if you extract all the properties and actions or methods that are not common to all tables, it seem that you might be left is an empty Form.

Empty (Photo credit: joshwept)

That may seem to be a disadvantage of this approach, but it is a strength. What sort of object do you get if you have an object which contains two box objects and a plank object? If you add the action or method of “eat off” you have a “dining table object” possibly with a property of “make-shift”.

So, under this extended Platonian scheme, what actually makes a table object a table object and not some other type of object? That’s actually a lot harder question than the question of what is a table object. Evidently it is a more subjective question as people will disagree what constitutes a table.

table with chairs
table with chairs (Photo credit: srqpix)

Cauliflower cheese

I found a recipe on the “Strands of my life” blog for Jamie Olivers’ 30 minute Cauliflower Macaroni Cheese and I decided to make a variation of it without the Macaroni. I didn’t try to do it in 30 minutes.

(Note: the recipe is lifted from his book, to which there is a link on the blog. Of course, you don’t have to buy the book to use the recipe, so the ethics of posting a recipe from a book might be a bit dodgy. On the other hand, the blog entry does act as an advertisement for the book.)

Jamie's 30 minute meals
Jamie’s 30 minute meals

Anyway, I did as the recipe suggests and put the bacon into the dish that I was going to use and put the dish in the oven, then started the cauliflower cooking. I actually needed to cut the cauliflower a little smaller than the recipe suggests as I was only cooking a small amount of cauliflower so used a smaller pan.

I grated the cheese, mixed it with the crème fraîche and some garlic powder. As the recipe directs, I took the dish with the bacon from the oven and processed the bacon with the bread and some rosemary leaves. I had expected the bacon to be crunchy, bit it was still soft, but I pressed on. The recipe calls for a little olive oil, which I added and which serves to cause the topping to crisp up nicely.

At this stage the cauliflower was cooked and I strained it, keeping the liquor as directed. Then it was just a matter of putting it together. The cauliflower went into the dish that the bacon had been cooking in. There was a fair bit of juice from the bacon in the dish, then in went some of the liquor from cooking the cauliflower, then the cheese/crème fraîche mixture. The mixture was a little too liquid, though the recipe calls for it to be “loose”, so next time I will add a little of the liquor first, then more after I see what it looks like.

Finally I covered the top with the bacon, breadcrumb and oil topping and cooked it as directed by the recipe. It came out of the oven like this. It could have been a little crisper on top.

Cauliflower cheese with bacon and breadcrumb topping

Here’s a picture of it on a plate. The mince was cooked the other day!

Cauliflower cheese and mince

Hoki and parsley fish cakes with Béchamel sauce

I had some left over mashed potatoes so I decided to get creative and make some fish cakes using some frozen hoki fillets. For added points I decided to make a Béchamel sauce. That’s a fancy name for a white sauce.

I started off by cooking the Hoki fillets in a little milk with a few herbs. (Did I mention that I’d thawed it first?) I used “Tuscan seasoning” by MasterFoods, which the label says contains salt, sugar, garlic, pepper, rosemary and parsley. The fish was cooked in about 15 minutes (at 180 degrees C) to the point where it flaked when poked with a fork.

Meanwhile I chopped some fresh parsley from the garden and added it to the mashed potato and stirred in two eggs. The mixture looked a little stiff so I added some of the liquid that the fish was cooked in. That was a mistake! I added too much and it went sloppy. Oh well. I added the cooked fish and put dollops of the mixture on a baking tray and cooked it at 200 degrees C for 20 minutes. “Dollops” is the right word. It was far too sloppy to form proper fish cakes.

I then made a roux. A “roux” is a fancy name for a 50-50 fat/flour mix which is cooked for a while. It is made into a white sauce (Béchamel) by slowly adding milk and cooking it (stirring all the time) until it is the desired consistency. I used the liquor that the fish was cooked in.

Here’s the “fish cakes”. I didn’t brush them with milk or egg, so they are a little palid. They are cooked though.

Fish “cakes”

This is what it looked like on my plate.

Plated up

Well, they tasted OK, but could have looked better! I’ve had enough runner beans for a while, but thankfully they seem to be taking a rest at the moment. Only one or two beans today.

Bangers and mash (with onion gravy)

I think I mentioned how to cook sausages in the oven the Jamie Oliver way before. Maybe not. Anyway the way to do it is to cook them on a wire rack. I didn’t have a suitable one so I made do with a roasting pan and the rack from another roasting pan that has long gone.

Makeshift sausage rack!

It didn’t quite fit but it worked. In summary Jamie’s method is to cook the sausages on 180 degrees C for 20 minutes, turning once.

The mash was no problem. I boiled the potatoes, drained them and mashed them with a little low fat milk and olive margarine. Of course, you could use full cream milk and butter if you wish!

The onion gravy was a new venture for me, but there are plenty of recipes on the Internet. I referred to this one. The basic idea is to cook the onions on a low heat, then add stock and then thicken as required. You have to watch the onions carefully, even on a low heat to ensure that they don’t caramelise. Of course a little colour doesn’t hurt. I put too much thickening in and the result was paler than I’d prefer, but it still tasted good. I used flour and water, so I had to cook the gravy a bit. If you use cornflower it isn’t necessary, according to the Internet anyway. I’ve not tried it.

Bangers and mash with onion gravy


Math Class
Math Class (Photo credit: attercop311)

On and off over the years I’ve been thinking about the concept of understanding, most particularly in the field of mathematics. When one encounters a new branch of mathematics, one is (conceptually) presented with the definitions, axioms and theorems in the field. Of course, one doesn’t necessarily understand them at first. However, after delving into the field, studying the axioms and definitions, and following through the proofs one soon gets the general feel for the field.

I might for instance read a ‘popular’ book about mathematics and read about ‘hyperreal numbers‘. By a ‘popular’ book, about maths, science or practically anything else, I mean a book which provides a non-rigorous introduction to a field, intended to give a feeling for it. If a reader of such a book requires an in-depth treatment of the subject, the reader would no doubt have to spend a long period studying the necessary maths to even start to address the topic. Such a ‘popular’ book could not possibly give one an understanding of ‘hyperreal numbers‘ of course. It’s more of a geography lesson, in that it would provide an understanding of where the topic fits in to the topography of mathematics, just as one might read about Greece, but one would not understand from the guide book what Greece is really like. For that one must go there.

Tired of studying!
Understanding mathematics requires study!

So you need to study something in maths to understand it. When do you know that you have understood it? I believe that there are at least three stages or signs that you understand it.

In the first stage, you have studied the axioms or the basis for the field of mathematics that you are trying to understand and you have seen and understood a few of the proofs of theorems in the field and you can reproduce them if asked. You have arrived in Greece, you are getting to grips with the money, you know how to purchase something in a cafe or restaurant, to extend the analogy.

In the second stage, you understand the ‘why’ of the proofs, you start to get a feel for the way that proofs are put together in the field and what can be achieved in the field. The concepts are sinking in. In Greece, you know how to find the best bars and restaurants where the Greeks drink and eat, you know what to order from the menus, and you are picking up a smattering of the language.

In the third and final stage, you are using the mathematics in the field with confidence, either to develop your own ideas in the field or in further study. When you look back at the difficulties you used to have in the field you are astonished that it took so long for the concepts to sink in. They seem obvious now. In the Greece analogy, you speak the language fluently, you have married a Greek person and you have lived in Greece for years. You mock the foreign tourists, and also you realise that there is more to Greece than your particular corner of it.

Greece (Photo credit: robynejay)

So, my point is that to understand something you need to immerse yourself in the topic. I don’t mean to say that you won’t have “Eureka Moments“, but even Archimedes had to immerse himself in his topic of study to come up with his Principle!

archimedes (Photo credit: Sputnik Beanburger III)

Cheese straws

OK, I ran out of snacks for Hamish and Duncan so I decided to try some cheese straws. I used the recipe here. The Guardian’s recipe promised “perfect Cheese straws”. Sounded good. The article actually goes into the topic of cheese straws in some depth quoting various chefs, but has its recipe towards the bottom of the main article, above the comments.

I made half quantities again, but even so, it only took a few minutes before my hands started to ache as I rubbed the butter in. I abandoned the manual approach and used a food mixer. This was the first time I’d seriously used a food mixer, so I’m not sure if I used the right blades or not. They were sharp-edged metal things.

I discovered a couple of things about processing stuff in food mixers.

Firstly, if you have to add a lot of an ingredient, a quantity of cheese for example, it’s a lot easier to add bulk ingredients by stopping the mixer and taking the top off and dumping it all in. Shoving it through the tiny hole at the top doesn’t really hack it! Doh!

Secondly, the recipe says to add water slowly until the dough ‘comes together in a firm dough’. What this means is that the dough suddenly goes from a breadcrumb-like consistency to a single sizeable lump and the mixer leaps about the work surface.

Anyway, here’s the dough in cling film ready for the fridge.

The dough ready for putting into the fridge for 30 minutes

After the dough had been in the fridge for 30 minutes, it was just a matter of making the straws and cooking them.

The dough rolled out

Making them was simple enough, but I misread the instructions and cooked them for only 5 minutes instead of 20 minutes! So I put them in for 5 minutes at a time until they were cooked. Actually they tasted a bit doughy so perhaps I should have cooked them longer. Here’s the finished product.

The end result
The end result. They are a little pallid as I misread the recipe!

The Pope thing and predestination.

I can understand the Pope, who is old, sick, and reportedly tired of the job, resigning. I can see that it has come as a surprise to many people. What I can’t understand is the reactions that people have to the announcement. It isn’t unprecedented, as Popes have resigned before.

Pope Benedict XVI
Pope Benedict XVI

However, some people believe that the Pope is elected for life and that resigning as Benedict XVI has done is close to sacrilege. I believe that the thinking goes something like this: God has chosen a particular man to be Pope. For that man to relinquish the post is for that man to, by his actions, imply that God is mistaken, and that cannot be.

If there is a God, and I’m firmly in the other camp, then He, being omnipotent and omniscient, not to mention omni-temporal, must know that His chosen candidate will resign the post, and has arranged matters so that this happens. In other words the resignation was ordained. It seems that one cannot get away from predestination! (The above description of God ‘knowing’ and ‘arranging’ is of course anthropomorphic).

pope and me
Someone similar in appearance from the back to the Pope contemplates the works of God or nature.

Put into a religious context, the debate over predestination versus free will comes down to the following:

If God is all-seeing and all powerful then He has control over everything and nothing happens which He hasn’t caused to happen. If there were something that He hadn’t caused to happen, then that would have to (de facto) be caused externally to God. God and whatever outside of God that caused the event would have to exist in much the same sense and such an existence would have to have a frame of existence, a ‘container’ if you like, that contains both God and the other thing. However, I started out by crediting God as being all-seeing and all powerful and that doesn’t seem to leave room for things outside of God as that implies something greater than God. So given the concept of an all powerful God there can be nothing that He hasn’t caused to happen and which he does not know about.

OK, so if God causes everything, then he causes every event to happen. He (ultimately) causes us to make the particular decisions that we make. So the concept of free will evaporates, as God causes us pick the options that we do, even if we think that we make a choice.

God, the Father watches us all everywhere.
God, the Father watches us all everywhere.

Of course God may be a compatibilist – He may believe that if he presents us with a number of options, we have the free will to select an option, even though the selection is in fact predetermined by Him.

Thomas Hobbes was a classical compatibilist
Thomas Hobbes was a classical compatibilist

Of course, shorn of the religious tones, the above argument still applies, if, for example, you replace the references to God with references to ‘Nature’ or ‘the natural laws’.


Damper or camp bread is a quick and easy bread which does not use yeast as a raising agent. Since it does not use yeast it needs either ‘self-raising’ flour or rising agents like baking powder.

The reason I was considering it was that I found that I’d forgotten to bake any bread. I decided to make some damper from a recipe that I found on the Internet. I chose to make half-quantities and may have miscalculated somewhere as the dough was dry and flakey even after adding all the specified liquid. (The recipe referred to uses water – some recipes use milk).

I added a little more water to get a reasonable dough and cooked it as specified. It came out a bit ‘blond’ probably because I didn’t glaze it with a little milk. It was very crumbly. I’ve not made or had damper before so I’m not sure if this is usual. The taste was more like scone than bread. Anyway, here it is.

Damper or camp bread. It is made without yeast.

Android app

I discovered an Android app for WordPress, so I’m giving it a try. (I may remove this post later).

The trouble is I don’t yet have a topic. Well, I haven’t yet posted any photographs, apart from the ones that I’ve used as illustrations.

Back lit tree fern frond.
I’m not good on plant names. This might be an azalea.
Grass. I was trying to capture the shape of the grass in the rather long lawn.
Grass and weeds. At this level the lawn seemed to be a battleground of competing species.