Though the answer may be integration, we don’t always know what the question is.

Similarly, though we often say “integration”, it’s not always clear what type of integration we mean. There are at least four interpretations of what we meant when we talk about “integration”:

- Integration across any of primary, secondary and tertiary healthcare
- Integration across health and social care (and education and housing and etc.) boundaries
- Integration of resources and processes
- Integration at the level of the individual.

As a mathematician by training, integration has another particular meaning to me. I thought it would be useful to reflect on what integration means from a mathematician’s perspective and so what we might learn from this in the context of health and social care.Mathematically, integration is the reverse process of differentiation. Differentiation is all about rates of change across different variables in a system. Differentiation is a way of thinking about the world as a result of combining infinitesimally small changes at particular points in time or space.

Integration, on the other hand, gives you a bigger sense of the whole. It tells you not just about rates of change but the overall picture you have: the sum total of what exists in time or space.

Differentiation is easier. It’s exciting (think Mick Jagger swaggering around a stage) and has no room for anything but the most important stuff. If there are any ‘spare’ numbers floating around then the process of differentiation gets rids of them – they disappear.

Integration, as any mathematician will tell you, is far harder. It’s a slower, altogether more considered process that requires more sophistication (think Bjork). There are some tricks you can use to make it slightly easier – such as integration by parts – but the challenge of integration remains.

And because integration is the reverse of differentiation it *adds in* an unknown factor: the arbitrary constant (from which this blog takes its name). Where differentiation has no space or time for the arbitrary constant, integration very deliberately includes it and recognises it. This unknown factor – an unidentified ingredient – is a vital component of integration.

(Interestingly, the only time the added, unknown ingredient of the arbitrary constant doesn’t play a part in integration is if you explicitly define the boundaries within which integration happens. By specifying these limits so exactly the arbitrary constant is cancelled out.)

If we were therefore to try and summarise what we know about integration from a mathematical point of view we’d say something like this:

- Integration is harder than differentiation – though there are limited tricks to make it easier
- It gives a bigger picture across a wider area than a specific view of just one point in time or space
- It has a secret ingredient – the arbitrary constant – which his fundamental to capturing this bigger picture
- This secret ingredient disappears only if you define exactly the boundaries of what integration is trying to achieve
- Integration is a subtle, complex process that takes time and understanding to do.

Thus, though you wouldn’t immediately think it, the mathematical conception of integration tells us everything we need to know about successful integration in public services, especially across health and social care and beyond.