I thank the noble and gallant Lord, Lord Craig of Radley, and the noble Viscount, Lord Hanworth, for tabling Amendments 12 to 14, and those who contributed to the debate. We recognise the fundamental importance of pure and applied maths to other sciences, and as the focus of scientific inquiry in its own right. It is right that we take the opportunity to note that importance here.

The noble and gallant Lord gave a number of potent examples of the importance of mathematical contributions to scientific innovation. Much like, we hope, the projects and advances that will be supported by ARIA, breakthroughs in mathematics can lead to unexpected leaps of progress in separate fields or find application in solving intractable and seemingly unrelated problems in other areas of science. As we just heard from the noble Lord, Lord Clement-Jones, who rightly reminded us, the UK has been home to many outstanding mathematicians of global significance, from Isaac Newton to Andrew Wiles.

However, I emphasise to the noble and gallant Lord, Lord Craig of Radley, and the noble Viscount, Lord Hanworth, that the drafting of the Clause that they have sought to amend follows existing powers in the Science and Technology Act 1965, and the Higher Education and Research Act 2017. It is important that it does so. Research into mathematics, including pure mathematics, has been funded in the UK using those powers for over five decades. Maths research is funded by the Engineering and Physical Sciences Research Council—one of the research councils that make up UKRI. The EPSRC spends more than £200 million on this theme, which includes research into maths in areas from number theory to topology and artificial intelligence. It is clear that maths is included in the definition of sciences as currently included in the Bill.

The 2004 guidance referenced by the noble Viscount, Lord Stansgate, predates the Higher Education and Research Act, which makes it clear that maths is included in the definition of science as drafted in the Bill. There is no need to particularise the interpretation through these amendments. Indeed, it would clearly be undesirable to seek to list exhaustively every possible field of scientific inquiry within the Bill. Departing from the existing embedded way these powers to fund research, including in mathematics, are drawn would be unhelpful.

ARIA’s programme managers will set ambitious programme-level goals. Although we do not often expect programme-level goals to lie within pure mathematics, it is right to highlight that ARIA might need to draw on pure and applied maths to achieve those goals, given their importance within the new fields noble Lords highlighted. It is right that ARIA may fund research in those areas.

We are confident that any activities of this nature that ARIA will seek to pursue are covered by its functions, and that the results of scientific research will encompass the results of mathematical inquiry that might be needed by ARIA. ARIA’s supplementary powers provide further reassurance. When exercising its functions, such as funding a programme with a specific scientific objective, ARIA’s supplementary powers allow it to do whatever is necessary in support of that. It is therefore the case that any mathematical endeavours that ARIA needed to draw on for a programme—for example, in support of a particular objective for machine learning—could be funded under its supplementary powers as well.

On that basis, although the noble and gallant Lord and the noble Viscount have raised important points, I hope they will be satisfied that there is no need for their amendments and feel able not to press them.