Enter the maze

A mathematical model of the mind

How does the physical world influence the mind? It's a profound question central to human experience that has troubled philosophers through the centuries. We sense the physical world around us, through, for example, measuring light intensity in the eye, detecting changes in air pressure in the ear, detecting changes in pressure on our skin, detecting chemicals in the air in our nose or chemicals in our food with our tongue; but how do these physical stimulation become a feeling of weight, or of hearing words or of tasting delicious food? How does the measurement of the stimulus by the body become the experience of perception in the mind?

A pile of pebbles on a hand

We still don't know the full answer, but back in 1860 a German doctor called Ernst Heinrich Weber, working with a colleague Gustav Fechner, discovered something rather fascinating. There is a simple mathematical rule (equation) relating the strength of stimulation to the strength of the perception, and it works across a whole range of our senses. This rule is called Weber's Law, and it's one of the first examples of a mathematical model relating body to mind. It's also interesting that while it was actually Fechner who did the maths, he gave the law as a 'gift' to Weber whose name it still carries today.

A weighty psychophysics experiment

Weber carried out several classic experiments to help him devise the rule. He blindfolded a man and gave him a weight to hold. Slowly Weber added more weight to the man's hand, until the man indicated he could first feel a difference. The weight was the stimulus strength, and the ability of the man to notice the difference in weight was the measure of a change in his perception. What Weber found was that the amount of extra weight he could add until the man could just notice the difference depended on how much weight there was in the hand at the start. If the weight was say only 10g to start with adding 1g more was noticeable, if the starting weight was say 1Kg then an extra 1g added wasn't perceived. This type of experiment where you manipulate something in the real physical world and measure the perception caused in a person's mind is called psychophysics, and it was Weber and Fechner who started this whole field of research.

Say it with maths

In words Weber's law says the stronger the original stimulus, the larger the change you need to make to notice that any thing has changed. Words are always useful and Weber could describe his findings, but looking at the experimental data Fechner was able to find a wonderfully simple mathematic description as well. If we call the stimulus strength S, (for example here this would be the original weight in the hand), and we can just notice the change if we increase the strength by amount dS, (remember dS is the amount of extra weight we add when we notice it), then (dS/S) = k where k is a constant, a number you can work out from the experimental data.

I predict that...

If we measure the constant k for one particular weight (by taking the measurements and doing the division), we can then test if it is really the same for other weights. Better still we can make predictions to test. Suppose we do it for a low initial weight of 10 g where we can just notice when 1g extra is added. Weber's law says 1g/10g is a constant - here 0.1. Using this experimental constant, 0.1, we can predict how much we should be able to add to the hand (dS) if we started with 1kg (1000g). The law says (dS/1000) must equal 0.1, so changing the subject of the formula dS =0.1x1000 = 100g. Our mathematics has made a specific mind/body prediction we can now go and test. Just using descriptive words we couldn't achieve this useful ability.

Useful all round rule

Experimentally psychophysics researchers have found that, if you don't go to extremes, then Weber's law is a good predictor relating stimulus strength to perception. The law holds for weight, light brightness, sound loudness and even line length and has many applications in computing, for example in image displays, computer graphics and audio processing. Weber's law is all around us though you may not have noticed. Ask yourself this, why cant you see the stars in the daytime? They are shining just as brightly as at night. Why don't you notice the ticking of a clock in the noisy daytime when it's there in the silence of the night? It's all just Weber's law hard at work.